tableau.h

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#pragma once
 
#include <vector>
#include <string>
#include <unordered_map>
#include "topology/operator.h"
 
namespace dg{
 
 
template<class real_type>
struct ButcherTableau{
    using value_type = real_type;
    ButcherTableau() = default;
    ButcherTableau(unsigned s, unsigned order,
                   const real_type* a , const real_type* b , const real_type* c):
        m_a(a, a+s*s), m_b(b, b+s), m_c(c, c+s), m_bt(b,b+s), m_q(order), m_p(order), m_s(s){}
    ButcherTableau(unsigned s, unsigned embedded_order, unsigned order,
               const real_type* a, const real_type* b, const real_type* bt, const real_type* c):
        m_a(a, a+s*s), m_b(b,b+s), m_c(c,c+s), m_bt(bt, bt+s), m_q(order), m_p(embedded_order), m_s(s), m_embedded(true){}
 
    ButcherTableau(unsigned s, real_type* data):
        m_a(s), m_b(s), m_c(s), m_bt(s), m_s(s), m_embedded(true)
    {
        for( unsigned i=0; i<s; i++)
        {
            m_c[i] = data[i*(s+1)];
            for( unsigned j=0; j<s; j++)
                m_a(i,j) = data[i*(s+1)+j+1];
        }
        m_q = (unsigned)data[s*(s+1)];
        for( unsigned j=0; j<s; j++)
            m_b[j] = data[s*(s+1)+j+1];
        m_p = (unsigned)data[(s+1)*(s+1)];
        for( unsigned j=0; j<s; j++)
            m_bt[j] = data[(s+1)*(s+1)+j+1];
    }
 
    real_type a( unsigned i, unsigned j) const {
        return m_a(i,j);
    }
    real_type c( unsigned i) const {
        return m_c[i];
    }
    real_type b( unsigned j) const {
        return m_b[j];
    }
    real_type bt( unsigned j) const {
        return m_bt[j];
    }
    real_type d( unsigned j) const {
        return m_b[j] - m_bt[j];
    }
    unsigned num_stages() const  {
        return m_s;
    }
    unsigned order() const {
        return m_q;
    }
    unsigned embedded_order() const{
        return m_p;
    }
    bool isEmbedded()const{
        return m_embedded;
    }
    bool isImplicit()const{
        for( unsigned i=0; i<m_s; i++)
            for( unsigned j=i; j<m_s; j++)
                if( a(i,j) != 0)
                    return true;
        return false;
    }
    bool isFsal()const{
        if( m_c[m_s-1] != 1)
            return false;
        for (unsigned j=0; j<m_s; j++)
            if( a(m_s-1,j) != b(j) )
                return false;
        return true;
    }
    private:
    dg::Operator<real_type> m_a;
    std::vector<real_type> m_b, m_c, m_bt;
    unsigned m_q, m_p, m_s;
    bool m_embedded = false;
};
 
template<class real_type>
struct ShuOsherTableau
{
    using value_type = real_type;
    ShuOsherTableau() = default;
    ShuOsherTableau( unsigned stages, unsigned order, const std::vector<real_type>& alpha_v, const std::vector<real_type>& beta_v){
        // parse the input into matrices
        m_stages = stages;
        m_order = order;
        m_alpha = dg::Operator<real_type>(stages, 0);
        m_beta = dg::Operator<real_type>(stages, 0);
        unsigned j=0;
        for( unsigned i=0; i<stages; i++)
            for( unsigned k=0; k<i+1; k++)
            {
                m_alpha(i,k) = alpha_v[j];
                m_beta(i,k) = beta_v[j];
                j++;
            }
        //std::cout << m_alpha << m_beta;
    }
 
    operator ButcherTableau<real_type>( )const{
        //the converse is not unique:
        //AN  EXTENSION  AND  ANALYSIS OF  THE
        //SHU-OSHER  REPRESENTATION OF  RUNGE-KUTTA  METHODS
        dg::Operator<real_type> a(m_stages, 0);
        std::vector< real_type > b(m_stages), c(m_stages);
        //convert to Butcher tableau
        for( unsigned i=1; i<m_stages; i++)
        {
            for( unsigned k=0; k<i; k++)
            {
                a( i, k) = m_beta ( i-1, k);
                for( unsigned j = k+1; j<i; j++)
                    a(i,k) += m_alpha( i-1,j)*a(j,k);
            }
        }
        for( unsigned k=0; k<m_stages; k++)
        {
            b[k] = m_beta( m_stages-1, k);
            for( unsigned j=k+1; j<m_stages; j++)
                b[k] += m_alpha(m_stages-1, j)*a( j, k);
        }
        for( unsigned i=0; i<m_stages; i++)
        {
            c[i] = a(i,0);
            for( unsigned k=1; k<m_stages; k++)
                c[i] += a(i,k);
        }
        dg::ButcherTableau<real_type> tableau(m_stages, m_order, &a.data()[0], &b[0], &c[0]);
        return tableau;
    }
    real_type alpha( unsigned i, unsigned j){ return m_alpha(i,j);}
    real_type beta( unsigned i, unsigned j){ return m_beta(i,j);}
    unsigned num_stages() const  {
        return m_stages;
    }
    unsigned order() const {
        return m_order;
    }
    private:
    unsigned m_stages, m_order;
    dg::Operator<real_type> m_alpha, m_beta;
};
namespace tableau{
//https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods
template<class real_type>
ButcherTableau<real_type> explicit_euler_1_1()
{
    real_type a[1] = {0};
    real_type b[1] = {1};
    real_type c[1] = {0};
    return ButcherTableau<real_type>( 1,1, a,b,c);
}
template<class real_type>
ButcherTableau<real_type> midpoint_2_2()
{
    real_type a[4] = {   0, 0,
                       0.5, 0 };
    real_type b[2] = {0., 1.};
    real_type c[2] = {0, 0.5};
    return ButcherTableau<real_type>( 2,2, a,b,c);
}
template<class real_type>
ButcherTableau<real_type> kutta_3_3()
{
    real_type a[9] = {0,0,0,
         .5, 0., 0.,
        -1., 2., 0.};
    real_type b[3] = {1./6., 2./3., 1./6.};
    real_type c[3] = {0, 0.5, 1.};
    return ButcherTableau<real_type>( 3,3, a,b,c);
}
template<class real_type>
ButcherTableau<real_type> classic_4_4()
{
    real_type a[16] = {
        0,0,0,0,
        0.5, 0,0,0,
        0,0.5,0,0,
        0,0,1,0
    };
    real_type b[4] = {1./6., 1./3., 1./3., 1./6.};
    real_type c[4] = {0, 0.5, 0.5, 1.};
    return ButcherTableau<real_type>( 4,4, a,b,c);
}
//From Yoh and Zhong (AIAA 42, 2004)
// !Attention! assumes another form of implementation
//than ARK tableaus
template<class real_type>
ButcherTableau<real_type> sirk3a_ex_3_3()
{
    real_type a[9] = {
        0,0,0,
        8./7., 0,0,
        71./252., 7./36.,0
    };
    real_type b[3] = {1./8., 1./8., 3./4.};
    real_type c[3] = {0, 8./7., 120./252.};
    return ButcherTableau<real_type>( 3,3, a,b,c);
}
template<class real_type>
ButcherTableau<real_type> sirk3a_im_3_3()
{
    real_type a[9] = {
        3./4.,0,0,
        5589./6524., 75./233.,0,
        7691./26096., -26335./78288., 65./168.
    };
    real_type b[3] = {1./8., 1./8., 3./4.};
    real_type c[3] = {3./4., 7689./6524., 27028./78288.};
    return ButcherTableau<real_type>( 3,3, a,b,c);
}
 
//tables copied from: https://sundials.readthedocs.io/en/latest/arkode/Butcher_link.html
template<class real_type>
ButcherTableau<real_type> heun_euler_2_1_2()
{
    real_type a[2*2] = {0,0, 1,0};
    real_type b[2] = {0.5, 0.5};
    real_type bt[2] = {1., 0.};
    real_type c[2] = {0,1};
    return ButcherTableau<real_type>(2,1,2, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> cavaglieri_exp_3_1_2()
{
    real_type a[3*3] = {
        0,0,0,
        2./5.,0, 0.,
        0., 1., 0.};
    real_type b[3] = {0., 5./6., 1./6.};
    real_type bt[3] = {0., 4./5., 1./5.};
    real_type c[3] = {0.,2./5., 1.};
    return ButcherTableau<real_type>(3,1,2, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> fehlberg_3_2_3()
{
    real_type a[3*3] = {
        0,0,0,
        1.,0,0,
        0.25, 0.25, 0.};
    real_type b[3] = {1./2., 1./2., 0.};
    real_type bt[3] = {1./6., 1./6., 2./3.};
    real_type c[3] = {0.,1., 0.5};
    return ButcherTableau<real_type>(3,2,3, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> fehlberg_4_2_3()
{
    real_type a[4*4] = {
        0,0,0,0,
        0.25,0,0,0,
        -189./800.,729./800.,0,0,
        214./891., 1./33., 650./891., 0.};
    real_type b[4] = {214./891., 1./33., 650./891., 0.};
    real_type bt[4] = {41./162., 0.,800./1053.,-1./78.};
    real_type c[4] = {0.,0.25,27./40.,1.0};
    return ButcherTableau<real_type>(4,2,3, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> bogacki_shampine_4_2_3()
{
    real_type a[4*4] = {
        0,0,0,0,
        0.5,0,0,0,
        0,0.75,0,0,
        2./9., 1./3., 4./9., 0.};
    real_type b[4] = {2./9., 1./3., 4./9., 0.};
    real_type bt[4] = {7./24., 1./4.,1./3.,1./8.};
    real_type c[4] = {0.,0.5,3./4.,1.};
    return ButcherTableau<real_type>(4,2,3, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> cavaglieri_exp_4_2_3()
{
    // IMEXRKCB3c
    real_type b[4] = {0., 673488652607./2334033219546., 493801219040./853653026979., 184814777513./1389668723319.};
    real_type c[4] = {0., 3375509829940./4525919076317., 272778623835./1039454778728., 1.0 };
    real_type a[4*4] = {
        0,0,0,0,
        c[1],0, 0.,0,
        b[0] , c[2], 0.,0.,
        b[0],b[1],c[3]-b[0]-b[1],0};
    real_type bt[4] = { 449556814708./1155810555193, 0., 210901428686./1400818478499., 480175564215./1042748212601.};
    return ButcherTableau<real_type>(4,2,3, a, b, bt, c);
    //real_type b[4] = {
    //-2179897048956./603118880443.,99189146040./891495457793.,6064140186914./1415701440113,146791865627./668377518349.
    //};
    //real_type c[4] = {0., 49./50., 1./25., 1.0 };
    //real_type a[4*4] = {
    //    0,0,0,0,
    //    c[1],0, 0.,0,
    //    13244205847./647648310246., 13419997131./686433909488., 0.,0.,
    //    b[0], 231677526244./1085522130027., 3007879347537./683461566472., 0};
    //real_type bt[4] = {
    //    0,0,25./48., 23./48.
    //};
    //return ButcherTableau<real_type>(4,2,3, a, b, bt, c);
 
}
template<class real_type>
ButcherTableau<real_type> ark324l2sa_erk_4_2_3()
{
    real_type data[5*6] = {
  0, 0, 0, 0, 0,
  1767732205903./2027836641118., 1767732205903./2027836641118., 0, 0, 0,
  3./5., 5535828885825./10492691773637., 788022342437./10882634858940., 0, 0,
  1, 6485989280629./16251701735622., -4246266847089./9704473918619., 10755448449292./10357097424841., 0. ,
  3, 1471266399579./7840856788654., -4482444167858./7529755066697., 11266239266428./11593286722821., 1767732205903./4055673282236.,
  2, 2756255671327./12835298489170., -10771552573575./22201958757719., 9247589265047./10645013368117., 2193209047091./5459859503100.
    };
    return ButcherTableau<real_type>(4,data);
}
template<class real_type>
ButcherTableau<real_type> zonneveld_5_3_4()
{
    real_type data[] = {
    0 , 0 , 0 , 0 , 0 , 0 ,
  1./2. , 1./2. , 0 , 0 , 0 , 0 ,
  1./2. , 0 , 1./2. , 0 , 0 , 0 ,
    1 , 0 , 0 , 1 , 0 , 0 ,
  3./4. , 5./32. , 7./32. , 13./32. , -1./32. , 0 ,
  4 , 1./6. , 1./3. , 1./3. , 1./6. , 0 ,
  3 , -1./2. , 7./3. , 7./3. , 13./6. , -16./3.};
    return ButcherTableau<real_type>(5,data);
}
template<class real_type>
ButcherTableau<real_type> ark436l2sa_erk_6_3_4()
{
    real_type data[] = {
    0 , 0 , 0 , 0 , 0 , 0 , 0 ,
    0.5 , 0.5 , 0 , 0 , 0 , 0 , 0 ,
    83./250. , 13861./62500. , 6889./62500. , 0 , 0 , 0 , 0 ,
    31./50. , -116923316275./2393684061468. , -2731218467317./15368042101831. , 9408046702089./11113171139209. , 0 , 0 , 0 ,
    17./20. , -451086348788./2902428689909. , -2682348792572./7519795681897. , 12662868775082./11960479115383. , 3355817975965./11060851509271. , 0 , 0 ,
    1 , 647845179188./3216320057751. , 73281519250./8382639484533. , 552539513391./3454668386233. , 3354512671639./8306763924573. , 4040./17871. , 0 ,
    4 , 82889./524892. , 0 , 15625./83664. , 69875./102672. , -2260./8211. , 0.25 ,
    3 , 4586570599./29645900160. , 0 , 178811875./945068544. , 814220225./1159782912. , -3700637./11593932. , 61727./225920.
    };
    return ButcherTableau<real_type>(6,data);
}
template<class real_type>
ButcherTableau<real_type> sayfy_aburub_6_3_4()
{
    real_type data[] = {
        0 , 0 , 0 , 0 , 0 , 0 , 0 ,
  1./2. , 1./2. , 0 , 0 , 0 , 0 , 0 ,
  1 , -1 , 2 , 0 , 0 , 0 , 0 ,
  1 , 1./6. , 2./3. , 1./6. , 0 , 0 , 0 ,
  1./2. , 0.137 , 0.226 , 0.137 , 0 , 0 , 0 ,
  1 , 0.452 , -0.904 , -0.548 , 0 , 2 , 0 ,
  4 , 1./6. , 1./3. , 1./12. , 0 , 1./3. , 1./12. ,
  3 , 1./6. , 2./3. , 1./6. , 0 , 0 , 0
    };
    return ButcherTableau<real_type>(6,data);
}
template<class real_type>
ButcherTableau<real_type> cash_karp_6_4_5()
{
    real_type data[] = {0 , 0 , 0 , 0 , 0 , 0 , 0 ,
  1./5. , 1./5. , 0 , 0 , 0 , 0 , 0 ,
  3./10. , 3./40. , 9./40. , 0 , 0 , 0 , 0 ,
  3./5. , 3./10. , -9./10. , 6./5. , 0 , 0 , 0 ,
  1 , -11./54. , 5./2. , -70./27. , 35./27. , 0 , 0 ,
  7./8. , 1631./55296. , 175./512. , 575./13824. , 44275./110592. , 253./4096. , 0 ,
  5 , 37./378. , 0 , 250./621. , 125./594. , 0 , 512./1771. ,
  4 , 2825./27648. , 0 , 18575./48384. , 13525./55296. , 277./14336. , 1./4.
  };
    return ButcherTableau<real_type>(6,data);
}
template<class real_type>
ButcherTableau<real_type> fehlberg_6_4_5()
{
    real_type data[] = {
        0 , 0 , 0 , 0 , 0 , 0 , 0 ,
  1./4. , 1./4. , 0 , 0 , 0 , 0 , 0 ,
  3./8. , 3./32. , 9./32. , 0 , 0 , 0 , 0 ,
  12./13. , 1932./2197. , -7200./2197. , 7296./2197. , 0 , 0 , 0 ,
  1 , 439./216. , -8 , 3680./513. , -845./4104. , 0 , 0 ,
  1./2. , -8./27. , 2 , -3544./2565. , 1859./4104. , -11./40. , 0 ,
  5 , 16./135. , 0 , 6656./12825. , 28561./56430. , -9./50. , 2./55. ,
  4 , 25./216. , 0 , 1408./2565. , 2197./4104. , -1./5. , 0
    };
    return ButcherTableau<real_type>(6,data);
}
template<class real_type>
ButcherTableau<real_type> dormand_prince_7_4_5()
{
    real_type data[] = {
        0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
  1./5. , 1./5. , 0 , 0 , 0 , 0 , 0 , 0 ,
  3./10. , 3./40. , 9./40. , 0 , 0 , 0 , 0 , 0 ,
  4./5. , 44./45. , -56./15. , 32./9. , 0 , 0 , 0 , 0 ,
  8./9. , 19372./6561. , -25360./2187. , 64448./6561. , -212./729. , 0 , 0 , 0 ,
  1 , 9017./3168. , -355./33. , 46732./5247. , 49./176. , -5103./18656. , 0 , 0 ,
  1 , 35./384. , 0 , 500./1113. , 125./192. , -2187./6784. , 11./84. , 0 ,
  5 , 35./384. , 0 , 500./1113. , 125./192. , -2187./6784. , 11./84. , 0 ,
  4 , 5179./57600. , 0 , 7571./16695. , 393./640. , -92097./339200. , 187./2100. , 1./40.
    };
    return ButcherTableau<real_type>(7,data);
}
template<class real_type>
ButcherTableau<real_type> tsitouras09_7_4_5()
{
    real_type b[7] = {
        0.091937670648056,1.156529958312496,-0.781330409541651,
        0.197624776163019,0.271639883438847,0.063598120979232,0
    };
    real_type bt[7] = {
        0.092167469090589,1.131750860603267,-0.759749304413104,
        0.205573577541223,0.264767065074229,0.040490332103796,1./40.
    };
    real_type c[7] = {
        0., 0.231572163526079, 0.212252555252816,0.596693497318054,
        0.797009955708112,1.,1.
    };
    real_type a[7*7] = {
        0,0,0,0,0,0,0,
        0,0,0,0,0,0,0,
        0,-0.059103796886580,0,0,0,0,0,
        0,4.560080615554683,-4.006458683473722,0,0,0,0,
        0,-2.443935658802774,2.631461258707441,0.524706566208284,0,0,0,
        0,9.516251378071800,-8.467630087008555,-0.987888827522473,0.867009765724064,0,0,
        0,0,0,0,0,0,0
    };
    for( unsigned i=0; i<6; i++)
    {
        real_type tmp=0;
        for( unsigned j=0; j<7; j++)
            tmp += a[i*7+j];
        a[i*7] = c[i] - tmp;
    }
    for( unsigned j=0; j<7; j++)
        a[6*7+j] = b[j];
    return ButcherTableau<real_type>(7,4,5,a,b,bt,c);
}
 
template<class real_type>
ButcherTableau<real_type> tsitouras11_7_4_5()
{
    // useful website
    //http://www.peterstone.name
    real_type b[7] = {
        0.09646076681806523,0.01,0.4798896504144996,
        1.379008574103742,-3.290069515436081,2.324710524099774,0.,
    };
    // There is a misprint in the original paper that prints b - bt instead of bt
    real_type bt[7] = {
        0.001780011052226,0.000816434459657,-0.007880878010262,0.144711007173263,-0.582357165452555,0.458082105929187,-1./66.
    };
    real_type c[7] = {
        0.,0.161,0.327,0.9,0.9800255409045097,1.,1.
    };
    real_type a[7*7] = {
        0,0,0,0,0,0,0,
        0,0,0,0,0,0,0,
        0,0.3354806554923570,0,0,0,0,0,
        0,-6.359448489975075,4.362295432869581,0,0,0,0,
        0,-11.74888356406283,7.495539342889836,-0.09249506636175525,0,0,0,
        0,-12.92096931784711,8.159367898576159,-0.07158497328140100,-0.02826905039406838,0,0,
        0,0,0,0,0,0,0
    };
    for( unsigned i=0; i<6; i++)
    {
        real_type tmp=0;
        for( unsigned j=0; j<7; j++)
            tmp += a[i*7+j];
        a[i*7] = c[i] - tmp;
    }
    for( unsigned j=0; j<7; j++)
    {
        a[6*7+j] = b[j];
        bt[j] = b[j] - bt[j];
    }
    return ButcherTableau<real_type>(7,4,5,a,b,bt,c);
}
 
template<class real_type>
ButcherTableau<real_type> ark548l2sa_erk_8_4_5()
{
    real_type data[] = {
0,0,0,0,0,0,0,0,0,
4./9.,4./9.,0,0,0,0,0,0,0,
6456083330201./8509243623797.,1./9.,1183333538310./1827251437969.,0,0,0,0,0,0,
1632083962415./14158861528103.,895379019517./9750411845327.,477606656805./13473228687314.,-112564739183./9373365219272.,0,0,0,0,0,
6365430648612./17842476412687.,-4458043123994./13015289567637.,-2500665203865./9342069639922.,983347055801./8893519644487.,2185051477207./2551468980502.,0,0,0,0,
18./25.,-167316361917./17121522574472.,1605541814917./7619724128744.,991021770328./13052792161721.,2342280609577./11279663441611.,3012424348531./12792462456678.,0,0,0,
191./200.,6680998715867./14310383562358.,5029118570809./3897454228471.,2415062538259./6382199904604.,-3924368632305./6964820224454.,-4331110370267./15021686902756.,-3944303808049./11994238218192.,0,0,
1,2193717860234./3570523412979.,2193717860234./3570523412979.,5952760925747./18750164281544.,-4412967128996./6196664114337.,4151782504231./36106512998704.,572599549169./6265429158920.,-457874356192./11306498036315.,0,
5,0,0,3517720773327./20256071687669.,4569610470461./17934693873752.,2819471173109./11655438449929.,3296210113763./10722700128969.,-1142099968913./5710983926999.,2./9.,
4,0,0,520639020421./8300446712847.,4550235134915./17827758688493.,1482366381361./6201654941325.,5551607622171./13911031047899.,-5266607656330./36788968843917.,1074053359553./5740751784926.
    };
 
    return ButcherTableau<real_type>( 8, data);
}
template<class real_type>
ButcherTableau<real_type> verner_9_5_6()
{
     /*
    coefficients copied from:
        http://people.math.sfu.ca/~jverner/
        http://people.math.sfu.ca/~jverner/RKV65.IIIXb.Efficient.00000144617.081204.CoeffsOnlyFLOAT
    */
    real_type c[9] = {0, 0.06, 0.09593333333333333333333333333333333333333, 0.1439, 0.4973, 0.9725, 0.9995, 1.0, 1.0};
    real_type a[9*9] = {
    0, 0,0,0,0,0,0,0,0,
    0.06, 0,0,0,0,0,0,0,0,
    0.01923996296296296296296296296296296296296, 0.07669337037037037037037037037037037037037,0,0,0,0,0,0,0,
     0.035975, 0, 0.107925, 0, 0, 0, 0, 0, 0,
     1.318683415233148260919747276431735612861, 0, -5.042058063628562225427761634715637693344, 4.220674648395413964508014358283902080483, 0, 0, 0, 0, 0,
     -41.87259166432751461803757780644346812905, 0, 159.4325621631374917700365669070346830453, -122.1192135650100309202516203389242140663, 5.531743066200053768252631238332999150076, 0, 0, 0, 0,
     -54.43015693531650433250642051294142461271, 0, 207.0672513650184644273657173866509835987, -158.6108137845899991828742424365058599469, 6.991816585950242321992597280791793907096, -0.01859723106220323397765171799549294623692, 0, 0, 0,
     -54.66374178728197680241215648050386959351, 0, 207.9528062553893734515824816699834244238, -159.2889574744995071508959805871426654216, 7.018743740796944434698170760964252490817, -0.01833878590504572306472782005141738268361, -0.0005119484997882099077875432497245168395840, 0, 0,
      0.03438957868357036009278820124728322386520, 0, 0, 0.2582624555633503404659558098586120858767, 0.4209371189673537150642551514069801967032, 4.405396469669310170148836816197095664891, -176.4831190242986576151740942499002125029, 172.3641334014150730294022582711902413315, 0};
 
    real_type b[9] = { 0.03438957868357036009278820124728322386520, 0, 0, 0.2582624555633503404659558098586120858767, 0.4209371189673537150642551514069801967032, 4.405396469669310170148836816197095664891, -176.4831190242986576151740942499002125029, 172.3641334014150730294022582711902413315, 0 };
    real_type bt[9] = { 0.04909967648382489730906854927971225836479, 0, 0, 0.2251112229516524153401395320539875329485, 0.4694682253029562039431948525047387412553, 0.8065792249988867707634161808995217981443, 0., -0.6071194891777959797672951465256217122488, 0.05686113944047569241147603178766138153594};
    return ButcherTableau<real_type>(9,5,6,a,b,bt,c);
}
template<class real_type>
ButcherTableau<real_type> verner_10_6_7()
{
   /*
    coefficients copied from:
        http://people.math.sfu.ca/~jverner/
        http://people.math.sfu.ca/~jverner/RKV76.IIa.Efficient.00001675585.081206.CoeffsOnlyFLOAT
    */
    real_type c[10] = { 0,  0.005, 0.1088888888888888888888888888888888888889, 0.1633333333333333333333333333333333333333, 0.4555000000000000000000000000000000000000, 0.6095094489978381317087004421486024949638, 0.884, 0.925, 1.0, 1.0};
    real_type a[10*10]={
        0, 0,0,0,0,0,0,0,0,0,
        0.005, 0,0,0,0,0,0,0,0,0,
     -1.076790123456790123456790123456790123457, 1.185679012345679012345679012345679012346, 0, 0, 0, 0, 0, 0, 0, 0,
    0.04083333333333333333333333333333333333333, 0, 0.1225, 0, 0, 0, 0, 0, 0, 0,
      0.6389139236255726780508121615993336109954, 0, -2.455672638223656809662640566430653894211, 2.272258714598084131611828404831320283215, 0, 0, 0, 0, 0, 0,
     -2.661577375018757131119259297861818119279, 0, 10.80451388645613769565396655365532838482, -8.353914657396199411968048547819291691541, 0.8204875949566569791420417341743839209619, 0, 0, 0, 0, 0,
      6.067741434696770992718360183877276714679, 0, -24.71127363591108579734203485290746001803, 20.42751793078889394045773111748346612697, -1.906157978816647150624096784352757010879, 1.006172249242068014790040335899474187268, 0, 0, 0, 0,
      12.05467007625320299509109452892778311648, 0, -49.75478495046898932807257615331444758322, 41.14288863860467663259698416710157354209, -4.461760149974004185641911603484815375051, 2.042334822239174959821717077708608543738, -0.09834843665406107379530801693870224403537, 0, 0, 0,
      10.13814652288180787641845141981689030769, 0, -42.64113603171750214622846006736635730625, 35.76384003992257007135021178023160054034, -4.348022840392907653340370296908245943710, 2.009862268377035895441943593011827554771, 0.3487490460338272405953822853053145879140, -0.2714390051048312842371587140910297407572, 0, 0,
     -45.03007203429867712435322405073769635151, 0, 187.3272437654588840752418206154201997384, -154.0288236935018690596728621034510402582, 18.56465306347536233859492332958439136765, -7.141809679295078854925420496823551192821, 1.308808578161378625114762706007696696508, 0, 0, 0};
    real_type b[10] = {  0.04715561848627222170431765108838175679569, 0, 0, 0.2575056429843415189596436101037687580986, 0.2621665397741262047713863095764527711129, 0.1521609265673855740323133199165117535523, 0.4939969170032484246907175893227876844296, -0.2943031171403250441557244744092703429139, 0.08131747232495109999734599440136761892478, 0};
    real_type bt[10] = { 0.04460860660634117628731817597479197781432, 0, 0, 0.2671640378571372680509102260943837899738, 0.2201018300177293019979715776650753096323, 0.2188431703143156830983120833512893824578, 0.2289871705411202883378173889763552365362, 0, 0, 0.02029518466335628222767054793810430358554};
    return ButcherTableau<real_type>(10,6,7, a,b,bt,c);
}
 
template<class real_type>
ButcherTableau<real_type> fehlberg_13_7_8()
{
    real_type data[] = {
        0,   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  2./27.,   2./27., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1./9.,   1./36., 1./12., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1./6.,   1./24., 0, 1./8., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  5./12.,   5./12., 0, -25./16., 25./16., 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1./2.,   1./20., 0, 0, 1./4., 1./5., 0, 0, 0, 0, 0, 0, 0, 0,
  5./6.,   -25./108., 0, 0, 125./108., -65./27., 125./54., 0, 0, 0, 0, 0, 0, 0,
  1./6.,   31./300., 0, 0, 0, 61./225., -2./9., 13./900., 0, 0, 0, 0, 0, 0,
  2./3.,   2, 0, 0, -53./6., 704./45., -107./9., 67./90., 3, 0, 0, 0, 0, 0,
  1./3.,   -91./108., 0, 0, 23./108., -976./135., 311./54., -19./60., 17./6., -1./12., 0, 0, 0, 0,
  1,   2383./4100., 0, 0, -341./164., 4496./1025., -301./82., 2133./4100., 45./82., 45./164., 18./41., 0, 0, 0,
  0,   3./205., 0, 0, 0, 0, -6./41., -3./205., -3./41., 3./41., 6./41., 0, 0, 0,
  1,   -1777./4100., 0, 0, -341./164., 4496./1025., -289./82., 2193./4100., 51./82., 33./164., 12./41., 0, 1, 0,
  8, 0, 0, 0, 0, 0, 34./105., 9./35., 9./35., 9./280., 9./280., 0, 41./840., 41./840. ,
  7, 41./840., 0, 0, 0, 0, 34./105., 9./35., 9./35., 9./280., 9./280., 41./840., 0, 0
    };
    return ButcherTableau<real_type>( 13, data);
}
template<class real_type>
ButcherTableau<real_type> dormand_prince_13_7_8()
{
    // only accurate to 18 significant figures
    // https://github.com/markmbaum/libode/blob/master/src/ode_dopri_87.cc
    real_type data[] = {
       0,   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       1.0/18, 1.0/18,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       1.0/12, 1.0/48, 1.0/16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       1.0/8,  1.0/32, 0, 3.0/32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       5.0/16, 5.0/16, 0,-75.0/64, 75.0/64, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       3.0/8,  3.0/80, 0,   0, 3.0/16, 3.0/20, 0, 0, 0, 0, 0, 0, 0, 0,
       59.0/400, 29443841.0/614563906,0,0,  77736538.0/692538347, -28693883.0/1125000000,  23124283.0/1800000000, 0,0,0,0,0,0,0,
       93.0/200, 16016141.0/946692911,0,0, 61564180.0/158732637, 22789713.0/633445777,  545815736.0/2771057229,  -180193667.0/1043307555, 0,0,0,0,0,0,
       5490023248.0/9719169821, 39632708.0/573591083,0,0, -433636366.0/683701615, -421739975.0/2616292301,  100302831.0/723423059, 790204164.0/839813087, 800635310.0/3783071287, 0,0,0,0,0,
       13.0/20,  246121993.0/1340847787, 0,0, -37695042795.0/15268766246,-309121744.0/1061227803, -12992083.0/490766935,  6005943493.0/2108947869, 393006217.0/1396673457,  123872331.0/1001029789, 0,0,0,0,
      1201146811.0/1299019798,  -1028468189.0/846180014, 0,0, 8478235783.0/508512852, 1311729495.0/1432422823, -10304129995.0/1701304382, -48777925059.0/3047939560, 15336726248.0/1032824649, -45442868181.0/3398467696,  3065993473.0/597172653, 0,0,0,
      1.0, 185892177.0/718116043, 0,0, -3185094517.0/667107341,-477755414.0/1098053517, -703635378.0/230739211,  5731566787.0/1027545527, 5232866602.0/850066563, -4093664535.0/808688257, 3962137247.0/1805957418, 65686358.0/487910083, 0,0,
      1.0, 403863854.0/491063109, 0,0, -5068492393.0/434740067, -411421997.0/543043805, 652783627.0/914296604, 11173962825.0/925320556, -13158990841.0/6184727034, 3936647629.0/1978049680, -160528059.0/685178525, 248638103.0/1413531060,0,0,
      8,  14005451.0/335480064,0,0, 0,0, -59238493.0/1068277825, 181606767.0/758867731,  561292985.0/797845732, -1041891430.0/1371343529, 760417239.0/1151165299,  118820643.0/751138087, -528747749.0/2220607170, 1.0/4,
      7,  13451932.0/455176632,0,0, 0,0, -808719846.0/976000145, 1757004468.0/5645159321, 656045339.0/265891186, -3867574721.0/1518517206, 465885868.0/322736535, 53011238.0/667516719, 2.0/45,0
    };
    return ButcherTableau<real_type>( 13, data);
}
//http://sce.uhcl.edu/rungekutta/
template<class real_type>
ButcherTableau<real_type> feagin_17_8_10()
{
    real_type a[17*17] = {
0,0,0,0,0, 0,0,0,0,0, 0,0,0,0,0, 0,0,
0.100000000000000000000000000000000000000000000000000000000000, 0,0,0,0, 0,0,0,0,0, 0,0,0,0,0, 0,0,
-0.915176561375291440520015019275342154318951387664369720564660,
 1.45453440217827322805250021715664459117622483736537873607016,0,0,0, 0,0,0,0,0, 0,0,0,0,0, 0,0,
0.202259190301118170324681949205488413821477543637878380814562,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.606777570903354510974045847616465241464432630913635142443687,0,0, 0,0,0,0,0, 0,0,0,0,0, 0,0,
0.184024714708643575149100693471120664216774047979591417844635,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.197966831227192369068141770510388793370637287463360401555746,
-0.0729547847313632629185146671595558023015011608914382961421311,0, 0,0,0,0,0, 0,0,0,0,0, 0,0,
 0.0879007340206681337319777094132125475918886824944548534041378,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.410459702520260645318174895920453426088035325902848695210406,
 0.482713753678866489204726942976896106809132737721421333413261, 0,0,0,0,0, 0,0,0,0,0, 0,0,
 0.0859700504902460302188480225945808401411132615636600222593880,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.330885963040722183948884057658753173648240154838402033448632,
 0.489662957309450192844507011135898201178015478433790097210790,
-0.0731856375070850736789057580558988816340355615025188195854775, 0,0,0,0, 0,0,0,0,0, 0,0,
 0.120930449125333720660378854927668953958938996999703678812621,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.260124675758295622809007617838335174368108756484693361887839,
 0.0325402621549091330158899334391231259332716675992700000776101,
-0.0595780211817361001560122202563305121444953672762930724538856, 0,0,0, 0,0,0,0,0, 0,0,
0.110854379580391483508936171010218441909425780168656559807038,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.0605761488255005587620924953655516875526344415354339234619466,
 0.321763705601778390100898799049878904081404368603077129251110,
 0.510485725608063031577759012285123416744672137031752354067590, 0,0, 0,0,0,0,0, 0,0,
0.112054414752879004829715002761802363003717611158172229329393,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.144942775902865915672349828340980777181668499748506838876185,
-0.333269719096256706589705211415746871709467423992115497968724,
 0.499269229556880061353316843969978567860276816592673201240332,
 0.509504608929686104236098690045386253986643232352989602185060, 0, 0,0,0,0,0, 0,0,
0.113976783964185986138004186736901163890724752541486831640341,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.0768813364203356938586214289120895270821349023390922987406384,
 0.239527360324390649107711455271882373019741311201004119339563,
 0.397774662368094639047830462488952104564716416343454639902613,
 0.0107558956873607455550609147441477450257136782823280838547024,
-0.327769124164018874147061087350233395378262992392394071906457, 0,0,0,0,0, 0,0,
0.0798314528280196046351426864486400322758737630423413945356284,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.0520329686800603076514949887612959068721311443881683526937298,
-0.0576954146168548881732784355283433509066159287152968723021864,
 0.194781915712104164976306262147382871156142921354409364738090,
 0.145384923188325069727524825977071194859203467568236523866582,
-0.0782942710351670777553986729725692447252077047239160551335016,
-0.114503299361098912184303164290554670970133218405658122674674, 0,0,0,0, 0,0,
0.985115610164857280120041500306517278413646677314195559520529,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.330885963040722183948884057658753173648240154838402033448632,
 0.489662957309450192844507011135898201178015478433790097210790,
-1.37896486574843567582112720930751902353904327148559471526397,
-0.861164195027635666673916999665534573351026060987427093314412,
 5.78428813637537220022999785486578436006872789689499172601856,
 3.28807761985103566890460615937314805477268252903342356581925,
-2.38633905093136384013422325215527866148401465975954104585807,
-3.25479342483643918654589367587788726747711504674780680269911,
-2.16343541686422982353954211300054820889678036420109999154887, 0,0,0, 0,0,
0.895080295771632891049613132336585138148156279241561345991710,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.197966831227192369068141770510388793370637287463360401555746,
-0.0729547847313632629185146671595558023015011608914382961421311,
 0.0000000000000000000000000000000000000000000000000000000000000,
-0.851236239662007619739049371445966793289359722875702227166105,
 0.398320112318533301719718614174373643336480918103773904231856,
 3.63937263181035606029412920047090044132027387893977804176229,
 1.54822877039830322365301663075174564919981736348973496313065,
-2.12221714704053716026062427460427261025318461146260124401561,
-1.58350398545326172713384349625753212757269188934434237975291,
-1.71561608285936264922031819751349098912615880827551992973034,
-0.0244036405750127452135415444412216875465593598370910566069132, 0,0, 0,0,
-0.915176561375291440520015019275342154318951387664369720564660,
 1.45453440217827322805250021715664459117622483736537873607016,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.777333643644968233538931228575302137803351053629547286334469,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.0910895662155176069593203555807484200111889091770101799647985,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.0910895662155176069593203555807484200111889091770101799647985,
 0.777333643644968233538931228575302137803351053629547286334469, 0, 0,0,
0.100000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
-0.157178665799771163367058998273128921867183754126709419409654,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.157178665799771163367058998273128921867183754126709419409654, 0,0,
0.181781300700095283888472062582262379650443831463199521664945,
 0.675000000000000000000000000000000000000000000000000000000000,
 0.342758159847189839942220553413850871742338734703958919937260,
 0.000000000000000000000000000000000000000000000000000000000000,
 0.259111214548322744512977076191767379267783684543182428778156,
-0.358278966717952089048961276721979397739750634673268802484271,
-1.04594895940883306095050068756409905131588123172378489286080,
 0.930327845415626983292300564432428777137601651182965794680397,
 1.77950959431708102446142106794824453926275743243327790536000,
 0.100000000000000000000000000000000000000000000000000000000000,
-0.282547569539044081612477785222287276408489375976211189952877,
-0.159327350119972549169261984373485859278031542127551931461821,
-0.145515894647001510860991961081084111308650130578626404945571,
-0.259111214548322744512977076191767379267783684543182428778156,
-0.342758159847189839942220553413850871742338734703958919937260,
-0.675000000000000000000000000000000000000000000000000000000000, 0
 
    };
    real_type b[17] = {
    0.0333333333333333333333333333333333333333333333333333333333333,
    0.0250000000000000000000000000000000000000000000000000000000000,
    0.0333333333333333333333333333333333333333333333333333333333333,
    0.000000000000000000000000000000000000000000000000000000000000,
    0.0500000000000000000000000000000000000000000000000000000000000,
    0.000000000000000000000000000000000000000000000000000000000000,
    0.0400000000000000000000000000000000000000000000000000000000000,
    0.000000000000000000000000000000000000000000000000000000000000,
    0.189237478148923490158306404106012326238162346948625830327194,
    0.277429188517743176508360262560654340428504319718040836339472,
    0.277429188517743176508360262560654340428504319718040836339472,
    0.189237478148923490158306404106012326238162346948625830327194,
    -0.0400000000000000000000000000000000000000000000000000000000000,
    -0.0500000000000000000000000000000000000000000000000000000000000,
    -0.0333333333333333333333333333333333333333333333333333333333333,
    -0.0250000000000000000000000000000000000000000000000000000000000,
    0.0333333333333333333333333333333333333333333333333333333333333
    };
    real_type bt[17] = {
    0.0333333333333333333333333333333333333333333333333333333333333,
    0.0277777777777777777777777777777777777777777777777777777777777,
    0.0333333333333333333333333333333333333333333333333333333333333,
    0.000000000000000000000000000000000000000000000000000000000000,
    0.0500000000000000000000000000000000000000000000000000000000000,
    0.000000000000000000000000000000000000000000000000000000000000,
    0.0400000000000000000000000000000000000000000000000000000000000,
    0.000000000000000000000000000000000000000000000000000000000000,
    0.189237478148923490158306404106012326238162346948625830327194,
    0.277429188517743176508360262560654340428504319718040836339472,
    0.277429188517743176508360262560654340428504319718040836339472,
    0.189237478148923490158306404106012326238162346948625830327194,
    -0.0400000000000000000000000000000000000000000000000000000000000,
    -0.0500000000000000000000000000000000000000000000000000000000000,
    -0.0333333333333333333333333333333333333333333333333333333333333,
    -0.0277777777777777777777777777777777777777777777777777777777777,
    0.0333333333333333333333333333333333333333333333333333333333333
    };
    real_type c[17] = {
 0.000000000000000000000000000000000000000000000000000000000000,
 0.100000000000000000000000000000000000000000000000000000000000,
 0.539357840802981787532485197881302436857273449701009015505500,
 0.809036761204472681298727796821953655285910174551513523258250,
 0.309036761204472681298727796821953655285910174551513523258250,
 0.981074190219795268254879548310562080489056746118724882027805,
 0.833333333333333333333333333333333333333333333333333333333333,
 0.354017365856802376329264185948796742115824053807373968324184,
 0.882527661964732346425501486979669075182867844268052119663791,
 0.642615758240322548157075497020439535959501736363212695909875,
 0.357384241759677451842924502979560464040498263636787304090125,
 0.117472338035267653574498513020330924817132155731947880336209,
 0.833333333333333333333333333333333333333333333333333333333333,
 0.309036761204472681298727796821953655285910174551513523258250,
 0.539357840802981787532485197881302436857273449701009015505500,
 0.100000000000000000000000000000000000000000000000000000000000,
 1.00000000000000000000000000000000000000000000000000000000000
    };
    return ButcherTableau<real_type>( 17, 8, 10, a, b,bt, c);
}
template<class real_type>
ButcherTableau<real_type> implicit_euler_1_1()
{
    real_type a[1] = {1};
    real_type b[1] = {1};
    real_type c[1] = {1};
    return ButcherTableau<real_type>( 1,1, a,b,c);
}
template<class real_type>
ButcherTableau<real_type> implicit_midpoint_1_2()
{
    real_type a[1] = { 0.5};
    real_type b[1] = { 1.};
    real_type c[1] = { 0.5};
    return ButcherTableau<real_type>( 1,2, a,b,c);
}
template<class real_type>
ButcherTableau<real_type> trapezoidal_2_2()
{
    real_type a[4] = { 0, 0, 0.5, 0.5};
    real_type b[2] = { 0.5, 0.5};
    real_type c[2] = { 0, 1.};
    return ButcherTableau<real_type>( 2,2, a,b,c);
}
 
template<class real_type>
ButcherTableau<real_type> sdirk_2_1_2()
{
    real_type x = (2.-sqrt(2.))/2.;
    real_type data[] = {
  x , x , 0 ,
  1.-x , 1.-2.*x , x,
  2 , 0.5 , 0.5,
  1 , 1 , 0
    };
    return ButcherTableau<real_type>( 2, data);
}
template<class real_type>
ButcherTableau<real_type> cavaglieri_imp_3_1_2()
{
    real_type a[3*3] = {
        0,0,0,
        0., 2./5., 0.,
        0., 5./6., 1./6.};
    real_type b[3] = {0., 5./6., 1./6.};
    real_type bt[3] = {0., 4./5., 1./5.};
    real_type c[3] = {0.,2./5., 1.};
    return ButcherTableau<real_type>(3,1,2, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> billington_3_3_2()
{
    real_type data[] = {
  0.292893218813 , 0.292893218813 , 0 , 0 ,
  1.091883092037 , 0.798989873223 , 0.292893218813 , 0 ,
  1.292893218813 , 0.740789228841 , 0.259210771159 , 0.292893218813 ,
  2 , 0.740789228840 , 0.259210771159 , 0 ,
  3 , 0.691665115992 , 0.503597029883 , -0.195262145876
    };
    return ButcherTableau<real_type>( 3, data);
}
template<class real_type>
ButcherTableau<real_type> trbdf2_3_3_2()
{
    real_type s2 = sqrt(2.);
    real_type data[] = {
      0, 0, 0, 0,
      2-s2, (2-s2)/2., (2-s2)/2., 0 ,
      1, s2/4., s2/4., (2-s2)/2.,
      2, s2/4., s2/4., (2-s2)/2.,
      3, (1-s2/4.)/3., (3*s2/4.+1.)/3., (2-s2)/6.
    };
    return ButcherTableau<real_type>( 3, data);
}
template<class real_type>
ButcherTableau<real_type> kvaerno_4_2_3()
{
    //Anne Kvaernoe
    real_type data[] = {
          0 , 0 , 0 , 0 , 0 ,
  0.871733043 , 0.4358665215 , 0.4358665215 , 0 , 0 ,
  1 , 0.490563388419108 , 0.073570090080892 , 0.4358665215 , 0 ,
  1 , 0.308809969973036 , 1.490563388254106 , -1.235239879727145 , 0.4358665215 ,
  3 , 0.308809969973036 , 1.490563388254106 , -1.235239879727145 , 0.4358665215 ,
  2 , 0.490563388419108 , 0.073570090080892 , 0.4358665215 , 0
    };
    return ButcherTableau<real_type>( 4, data);
}
template<class real_type>
ButcherTableau<real_type> sanchez_3_3()
{
    real_type a0 = 1.351207191959658;
    real_type b[3] = {a0,a0,1.-2.*a0};
    real_type a[3*3] = {
        b[0]/2.,0,0,
        b[0], b[1]/2., 0.,
        b[0], b[1], b[2]/2.};
    real_type c[3] = {b[0]/2.,b[0]+b[1]/2., b[0]+b[1]+b[2]/2.};
    return ButcherTableau<real_type>(3,4,a,b,c);
}
template<class real_type>
ButcherTableau<real_type> sanchez_3_4()
{
    real_type a0 = 1.351207191959658;
    real_type b[3] = {a0,1-2.*a0,a0};
    real_type a[3*3] = {
        b[0]/2.,0,0,
        b[0], b[1]/2., 0.,
        b[0], b[1], b[2]/2.};
    real_type c[3] = {b[0]/2.,b[0]+b[1]/2., b[0]+b[1]+b[2]/2.};
    return ButcherTableau<real_type>(3,4,a,b,c);
 
}
template<class real_type>
ButcherTableau<real_type> sanchez_6_5()
{
    real_type b[6] = {0.5080048194000274, 1.360107162294827,
        2.0192933591817224, 0.5685658926458251, -1.4598520495864393,
        -1.9961191839359627 };
    real_type a[6*6];
    real_type c[6] = {0,0,0,0,0,0};
    for( unsigned i=0; i<6; i++)
        for( unsigned k=0; k<6; k++)
        {
            if( k==i)
                a[i*6+k] = b[k]/2.;
            else if( k<i)
                a[i*6+k] = b[k];
            else
                a[i*6+k] = 0.;
            c[i]+=a[i*6+k];
        }
    return ButcherTableau<real_type>(6,5,a,b,c);
}
template<class real_type>
ButcherTableau<real_type> sanchez_7_6()
{
    real_type b[7] = {7.8451361047755652e-01, 2.3557321335935860e-01,
        -1.1776799841788705, 1.3151863206839107, -1.1776799841788705,
        2.3557321335935860e-01, 7.8451361047755652e-01 };
    real_type a[7*7];
    real_type c[7] = {0,0,0,0,0,0,0};
    for( unsigned i=0; i<7; i++)
        for( unsigned k=0; k<7; k++)
        {
            if( k==i)
                a[i*7+k] = b[k]/2.;
            else if( k<i)
                a[i*7+k] = b[k];
            else
                a[i*7+k] = 0.;
            c[i]+=a[i*7+k];
        }
    return ButcherTableau<real_type>(7,6,a,b,c);
}
template<class real_type>
ButcherTableau<real_type> sdirk_4_2_3()
{
    real_type data[] = {
        1./4. , 1./4. , 0 , 0 , 0 ,
  11./28. , 1./7. , 1./4. , 0 , 0 ,
  1./3. , 61./144. , -49./144. , 1./4. , 0 ,
  1. , 0. , 0. , 3./4. , 1./4. ,
  3 , 0. , 0. , 3./4. , 1./4. ,
  2 , -61./600., 49/600., 79./100. , 23./100.
    };
    return ButcherTableau<real_type>( 4, data);
}
 
template<class real_type>
ButcherTableau<real_type> cavaglieri_imp_4_2_3()
{
    // IMEXRKCB3c
    real_type b[4] = {0., 673488652607./2334033219546., 493801219040./853653026979., 184814777513./1389668723319.};
    real_type c[4] = {0., 3375509829940./4525919076317., 272778623835./1039454778728., 1.0 };
    real_type a[4*4] = {
        0,0,0,0,
        0., 3375509829940./4525919076317., 0.,0,
        b[0] , -11712383888607531889907./32694570495602105556248., 566138307881./912153721139., 0.,
        b[0],b[1], b[2],b[3]};
    real_type bt[4] = { 0., 366319659506./ 1093160237145., 270096253287./ 480244073137., 104228367309./ 1017021570740. };
    //real_type b[4] = {
    //-2179897048956./603118880443.,99189146040./891495457793.,6064140186914./1415701440113,146791865627./668377518349.
    //};
    //real_type c[4] = {0., 49./50., 1./25., 1.0 };
    //real_type a[4*4] = {
    //    0,0,0,0,
    //    c[1]/2.,c[1]/2., 0.,0,
    //    -785157464198./1093480182337.,-30736234873./978681420651.,983779726483./1246172347126.,0,
    //    b[0], b[1], b[2], b[3]};
    //real_type bt[4] = {
    //    0.,337712514207./759004992869.,311412265155./608745789881.,52826596233./1214539205236.
    //};
    return ButcherTableau<real_type>(4,2,3, a, b, bt, c);
}
template<class real_type>
ButcherTableau<real_type> ark324l2sa_dirk_4_2_3()
{
    real_type data[] = {
          0 , 0 , 0 , 0 , 0 ,
  1767732205903./2027836641118. , 1767732205903./4055673282236. , 1767732205903./4055673282236. , 0 , 0 ,
  3./5. , 2746238789719./10658868560708. , -640167445237./6845629431997. , 1767732205903./4055673282236. , 0 ,
  1 , 1471266399579./7840856788654. , -4482444167858./7529755066697. , 11266239266428./11593286722821. , 1767732205903./4055673282236. ,
  3 , 1471266399579./7840856788654. , -4482444167858./7529755066697. , 11266239266428./11593286722821. , 1767732205903./4055673282236. ,
  2 , 2756255671327./12835298489170. , -10771552573575./22201958757719. , 9247589265047./10645013368117. , 2193209047091./5459859503100.
    };
    return ButcherTableau<real_type>( 4, data);
}
template<class real_type>
ButcherTableau<real_type> cash_5_2_4()
{
    real_type data[] = {
          0.435866521508 , 0.435866521508 , 0 , 0 , 0 , 0 ,
  -0.7 , -1.13586652150 , 0.435866521508 , 0 , 0 , 0 ,
  0.8 , 1.08543330679 , -0.721299828287 , 0.435866521508 , 0 , 0 ,
  0.924556761814 , 0.416349501547 , 0.190984004184 , -0.118643265417 , 0.435866521508 , 0 ,
  1 , 0.896869652944 , 0.0182725272734 , -0.0845900310706 , -0.266418670647 , 0.435866521508 ,
  4 , 0.896869652944 , 0.0182725272734 , -0.0845900310706 , -0.266418670647 , 0.435866521508 ,
  2 , 1.05646216107052 , -0.0564621610705236 , 0 , 0 , 0
    };
    return ButcherTableau<real_type>( 5, data);
}
template<class real_type>
ButcherTableau<real_type> cash_5_3_4()
{
    real_type data[] = {
          0.435866521508 , 0.435866521508 , 0 , 0 , 0 , 0 ,
  -0.7 , -1.13586652150 , 0.435866521508 , 0 , 0 , 0 ,
  0.8 , 1.08543330679 , -0.721299828287 , 0.435866521508 , 0 , 0 ,
  0.924556761814 , 0.416349501547 , 0.190984004184 , -0.118643265417 , 0.435866521508 , 0 ,
  1 , 0.896869652944 , 0.0182725272734 , -0.0845900310706 , -0.266418670647 , 0.435866521508 ,
  4 , 0.896869652944 , 0.0182725272734 , -0.0845900310706 , -0.266418670647 , 0.435866521508 ,
  3 , 0.776691932910 , 0.0297472791484 , -0.0267440239074 , 0.220304811849 , 0
    };
    return ButcherTableau<real_type>( 5, data);
}
template<class real_type>
ButcherTableau<real_type> sdirk_5_3_4()
{
    real_type data[] = {
        1./4. , 1./4. , 0 , 0 , 0 , 0 ,
  3./4. , 1./2. , 1./4. , 0 , 0 , 0 ,
  11./20. , 17./50. , -1./25. , 1./4. , 0 , 0 ,
  1./2. , 371./1360. , -137./2720. , 15./544. , 1./4. , 0 ,
  1 , 25./24. , -49./48. , 125./16. , -85./12. , 1./4. ,
  4 , 25./24. , -49./48. , 125./16. , -85./12. , 1./4. ,
  3 , 59./48. , -17./96. , 225./32. , -85./12. , 0
    };
    return ButcherTableau<real_type>( 5, data);
}
template<class real_type>
ButcherTableau<real_type> kvaerno_5_3_4()
{
    //Anne Kvaernoe
    real_type data[] = {
        0 , 0 , 0 , 0 , 0 , 0 ,
  0.871733043 , 0.4358665215  , 0.4358665215  , 0 , 0 , 0 ,
  0.468238744853136 , 0.140737774731968 , -0.108365551378832 , 0.4358665215 , 0 , 0 ,
  1 , 0.102399400616089 , -0.376878452267324 , 0.838612530151233 , 0.4358665215 , 0 ,
  1 , 0.157024897860995 , 0.117330441357768 , 0.61667803039168 , -0.326899891110444 , 0.4358665215 ,
  4 , 0.157024897860995 , 0.117330441357768 , 0.61667803039168 , -0.326899891110444 , 0.4358665215 ,
  3 , 0.102399400616089 , -0.376878452267324 , 0.838612530151233 , 0.4358665215 , 0
    };
    return ButcherTableau<real_type>( 5, data);
}
template<class real_type>
ButcherTableau<real_type> ark436l2sa_dirk_6_3_4()
{
    real_type data[] = {
        0 , 0 , 0 , 0 , 0 , 0 , 0 ,
  1./2. , 1./4. , 1./4. , 0 , 0 , 0 , 0 ,
  83./250. , 8611./62500. , -1743./31250. , 1./4. , 0 , 0 , 0 ,
  31./50. , 5012029./34652500. , -654441./2922500. , 174375./388108. , 1./4. , 0 , 0 ,
  17./20. , 15267082809./155376265600. , -71443401./120774400. , 730878875./902184768. , 2285395./8070912. , 1./4. , 0 ,
  1 , 82889./524892. , 0 , 15625./83664. , 69875./102672. , -2260./8211. , 1./4. ,
  4 , 82889./524892. , 0 , 15625./83664. , 69875./102672. , -2260./8211. , 1./4. ,
  3 , 4586570599./29645900160. , 0 , 178811875./945068544. , 814220225./1159782912. , -3700637./11593932. , 61727./225920.
    };
    return ButcherTableau<real_type>( 6, data);
}
template<class real_type>
ButcherTableau<real_type> kvaerno_7_4_5()
{
    //Anne Kvaernoe
    real_type data[] = {
        0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
  0.52 , 0.26 , 0.26 , 0 , 0 , 0 , 0 , 0 ,
  1.230333209967908 , 0.13 , 0.84033320996790809 , 0.26 , 0 , 0 , 0 , 0 ,
  0.895765984350076 , 0.22371961478320505 , 0.47675532319799699 , -0.06470895363112615 , 0.26 , 0 , 0 , 0 ,
  0.436393609858648 , 0.16648564323248321 , 0.10450018841591720 , 0.03631482272098715 , -0.13090704451073998 , 0.26 , 0 , 0 ,
  1 , 0.13855640231268224 , 0 , -0.04245337201752043 , 0.02446657898003141 , 0.61943039072480676 , 0.26 , 0 ,
  1 , 0.13659751177640291 , 0 , -0.05496908796538376 , -0.04118626728321046 , 0.62993304899016403 , 0.06962479448202728 , 0.26 ,
  5 , 0.13659751177640291 , 0 , -0.05496908796538376 , -0.04118626728321046 , 0.62993304899016403 , 0.06962479448202728 , 0.26 ,
  4 , 0.13855640231268224 , 0 , -0.04245337201752043 , 0.02446657898003141 , 0.61943039072480676 , 0.26 , 0
    };
    return ButcherTableau<real_type>( 7, data);
}
template<class real_type>
ButcherTableau<real_type> ark548l2sa_dirk_8_4_5()
{
    real_type data[] = {
0,0,0,0,0,0,0,0,0,
4./9.,2./9.,2./9.,0,0,0,0,0,0,
6456083330201./8509243623797.,2366667076620./8822750406821.,2366667076620./8822750406821.,2./9.,0,0,0,0,0,
1632083962415./14158861528103.,-257962897183./4451812247028.,-257962897183./4451812247028.,128530224461./14379561246022.,2./9.,0,0,0,0,
6365430648612./17842476412687.,-486229321650./11227943450093.,-486229321650./11227943450093.,-225633144460./6633558740617.,1741320951451./6824444397158.,2./9.,0,0,0,
18./25.,621307788657./4714163060173.,621307788657./4714163060173.,-125196015625./3866852212004.,940440206406./7593089888465.,961109811699./6734810228204.,2./9.,0,0,
191./200.,2036305566805./6583108094622.,2036305566805./6583108094622.,-3039402635899./4450598839912.,-1829510709469./31102090912115.,-286320471013./6931253422520.,8651533662697./9642993110008.,2./9.,0,
1,0,0,3517720773327./20256071687669.,4569610470461./17934693873752.,2819471173109./11655438449929.,3296210113763./10722700128969.,-1142099968913./5710983926999.,2./9.,
5,0,0,3517720773327./20256071687669.,4569610470461./17934693873752.,2819471173109./11655438449929.,3296210113763./10722700128969.,-1142099968913./5710983926999.,2./9.,
4,0,0,520639020421./8300446712847.,4550235134915./17827758688493.,1482366381361./6201654941325.,5551607622171./13911031047899.,-5266607656330./36788968843917.,1074053359553./5740751784926.
    };
    return ButcherTableau<real_type>( 8, data);
}
// Convert between Shu-Osher representation to a Butcher tableau
template<class real_type>
dg::ButcherTableau<real_type> shuosher2butcher( unsigned stages, unsigned order, std::vector<real_type> alpha_v, std::vector<real_type> beta_v)
{
    ShuOsherTableau<real_type> shu( stages, order, alpha_v, beta_v);
    return dg::ButcherTableau<real_type>(shu);
}
 
template<class real_type>
ShuOsherTableau<real_type> ssprk_2_2()
{
    unsigned stages=2, order = 2;
    std::vector<double> alpha_v = {1., 0.5, 0.5};
    std::vector<double> beta_v = {1., 0., 0.5};
    return ShuOsherTableau<real_type>( stages, order, alpha_v, beta_v);
}
template<class real_type>
ShuOsherTableau<real_type> ssprk_3_2()
{
    //CFLL = 2 -> eff = 0.66
    unsigned stages=3, order = 2;
    std::vector<double> alpha_v = {1., 0, 1., 1./3., 0, 2./3.};
    std::vector<double> beta_v = {0.5, 0., 0.5, 0., 0., 1./3.};
    return ShuOsherTableau<real_type>( stages, order, alpha_v, beta_v);
}
template<class real_type>
ShuOsherTableau<real_type> ssprk_3_3()
{
    //CFL = 1 -> eff = 0.33
    unsigned stages=3, order = 3;
    std::vector<double> alpha_v = {1.,3./4.,1./4.,1./3.,0.,2./3.};
    std::vector<double> beta_v = {1., 0., 1./4.,0.,0.,2./3.};
    return ShuOsherTableau<real_type>( stages, order, alpha_v, beta_v);
}
template<class real_type>
ShuOsherTableau<real_type> ssprk_5_3()
{
    //Ruuth 2005
    //CFL = 2.6 -> eff = 0.5
    unsigned stages=5, order = 3;
    std::vector<double> alpha_v = {
        1, 0, 1, 0.56656131914033, 0, 0.43343868085967, 0.09299483444413, 0.00002090369620, 0, 0.90698426185967, 0.00736132260920, 0.20127980325145, 0.00182955389682, 0, 0.78952932024253
    };
    std::vector<double> beta_v = {
        0.37726891511710, 0, 0.37726891511710, 0, 0, 0.16352294089771, 0.00071997378654, 0, 0, 0.34217696850008, 0.00277719819460, 0.00001567934613, 0, 0, 0.29786487010104
    };
    return ShuOsherTableau<real_type>( stages, order, alpha_v, beta_v);
}
 
template<class real_type>
ShuOsherTableau<real_type> ssprk_5_4()
{
    //Spiteri & Ruuth 2005
    //CLM = 1.5 -> eff = 0.37 , better than 3_3
    unsigned stages=5, order = 4;
    std::vector<double> alpha_v = {
        1, 0.44437049406734, 0.55562950593266, 0.62010185138540, 0, 0.37989814861460, 0.17807995410773, 0, 0, 0.82192004589227, 0.00683325884039, 0, 0.51723167208978, 0.12759831133288, 0.34833675773694
    };
    std::vector<double> beta_v = {
        0.39175222700392, 0, 0.36841059262959, 0, 0, 0.25189177424738, 0, 0, 0, 0.54497475021237, 0, 0, 0, 0.08460416338212, 0.22600748319395
    };
    return ShuOsherTableau<real_type>( stages, order, alpha_v, beta_v);
}
 
 
}//namespace tableau
 
enum tableau_identifier{
    EXPLICIT_EULER_1_1, 
    MIDPOINT_2_2, 
    KUTTA_3_3,
    CLASSIC_4_4,
    //ARKode tableaus
    HEUN_EULER_2_1_2,
    CAVAGLIERI_3_1_2, 
    FEHLBERG_3_2_3,
    FEHLBERG_4_2_3,
    BOGACKI_SHAMPINE_4_2_3,
    CAVAGLIERI_4_2_3, 
    ARK324L2SA_ERK_4_2_3,
    ZONNEVELD_5_3_4,
    ARK436L2SA_ERK_6_3_4,
    SAYFY_ABURUB_6_3_4,
    CASH_KARP_6_4_5,
    FEHLBERG_6_4_5,
    DORMAND_PRINCE_7_4_5,
    TSITOURAS09_7_4_5,
    TSITOURAS11_7_4_5,
    ARK548L2SA_ERK_8_4_5,
    VERNER_9_5_6,
    VERNER_10_6_7,
    FEHLBERG_13_7_8,
    DORMAND_PRINCE_13_7_8,
    FEAGIN_17_8_10,
    //implicit ARKode tableaus
    IMPLICIT_EULER_1_1,
    IMPLICIT_MIDPOINT_1_2, 
    TRAPEZOIDAL_2_2,
    SDIRK_2_1_2, 
    CAVAGLIERI_IMPLICIT_3_1_2, 
    BILLINGTON_3_3_2,
    TRBDF2_3_3_2,
    SANCHEZ_3_3,
    KVAERNO_4_2_3,
    SDIRK_4_2_3,
    CAVAGLIERI_IMPLICIT_4_2_3, 
    ARK324L2SA_DIRK_4_2_3,
    SANCHEZ_3_4,
    CASH_5_2_4,
    CASH_5_3_4,
    SDIRK_5_3_4,
    KVAERNO_5_3_4,
    ARK436L2SA_DIRK_6_3_4,
    KVAERNO_7_4_5,
    SANCHEZ_6_5,
    ARK548L2SA_DIRK_8_4_5,
    SANCHEZ_7_6,
    // SSP RK tableaus
    SSPRK_2_2, 
    SSPRK_3_2, 
    SSPRK_3_3, 
    SSPRK_5_3, 
    SSPRK_5_4 
};
 
namespace create{
 
static std::unordered_map<std::string, enum tableau_identifier> str2id{
    //Explicit methods
    {"Euler", EXPLICIT_EULER_1_1},
    {"Midpoint-2-2", MIDPOINT_2_2},
    {"Kutta-3-3", KUTTA_3_3},
    {"Runge-Kutta-4-4", CLASSIC_4_4},
    //Embedded explicit methods
    {"Heun-Euler-2-1-2", HEUN_EULER_2_1_2},
    {"Cavaglieri-3-1-2 (explicit)", CAVAGLIERI_3_1_2},
    {"Fehlberg-3-2-3", FEHLBERG_3_2_3},
    {"Fehlberg-4-2-3", FEHLBERG_4_2_3},
    {"Bogacki-Shampine-4-2-3", BOGACKI_SHAMPINE_4_2_3},
    {"Cavaglieri-4-2-3 (explicit)", CAVAGLIERI_4_2_3},
    {"ARK-4-2-3 (explicit)", ARK324L2SA_ERK_4_2_3},
    {"Zonneveld-5-3-4", ZONNEVELD_5_3_4},
    {"ARK-6-3-4 (explicit)", ARK436L2SA_ERK_6_3_4},
    {"Sayfy-Aburub-6-3-4", SAYFY_ABURUB_6_3_4},
    {"Cash-Karp-6-4-5", CASH_KARP_6_4_5},
    {"Fehlberg-6-4-5", FEHLBERG_6_4_5},
    {"Dormand-Prince-7-4-5", DORMAND_PRINCE_7_4_5},
    {"Tsitouras09-7-4-5", TSITOURAS09_7_4_5},
    {"Tsitouras11-7-4-5", TSITOURAS11_7_4_5},
    {"ARK-8-4-5 (explicit)", ARK548L2SA_ERK_8_4_5},
    {"Verner-9-5-6", VERNER_9_5_6},
    {"Verner-10-6-7", VERNER_10_6_7},
    {"Fehlberg-13-7-8", FEHLBERG_13_7_8},
    {"Dormand-Prince-13-7-8", DORMAND_PRINCE_13_7_8},
    {"Feagin-17-8-10", FEAGIN_17_8_10},
    //Implicit methods
    {"Euler (implicit)", IMPLICIT_EULER_1_1},
    {"Midpoint (implicit)", IMPLICIT_MIDPOINT_1_2},
    {"Trapezoidal-2-2", TRAPEZOIDAL_2_2},
    {"SDIRK-2-1-2", SDIRK_2_1_2},
    {"Cavaglieri-3-1-2 (implicit)", CAVAGLIERI_IMPLICIT_3_1_2},
    {"Billington-3-3-2", BILLINGTON_3_3_2},
    {"TRBDF2-3-3-2", TRBDF2_3_3_2},
    {"Sanchez-3-3", SANCHEZ_3_3},
    {"Sanchez-3-4", SANCHEZ_3_4},
    {"Sanchez-6-5", SANCHEZ_6_5},
    {"Sanchez-7-6", SANCHEZ_7_6},
    {"Kvaerno-4-2-3", KVAERNO_4_2_3},
    {"SDIRK-4-2-3", SDIRK_4_2_3},
    {"Cavaglieri-4-2-3 (implicit)", CAVAGLIERI_IMPLICIT_4_2_3},
    {"ARK-4-2-3 (implicit)", ARK324L2SA_DIRK_4_2_3},
    {"Cash-5-2-4", CASH_5_2_4},
    {"Cash-5-3-4", CASH_5_3_4},
    {"SDIRK-5-3-4", SDIRK_5_3_4},
    {"Kvaerno-5-3-4", KVAERNO_5_3_4},
    {"ARK-6-3-4 (implicit)", ARK436L2SA_DIRK_6_3_4},
    {"Kvaerno-7-4-5", KVAERNO_7_4_5},
    {"ARK-8-4-5 (implicit)", ARK548L2SA_DIRK_8_4_5},
    //Shu-Osher methods
    {"SSPRK-2-2", SSPRK_2_2},
    {"SSPRK-3-2", SSPRK_3_2},
    {"SSPRK-3-3", SSPRK_3_3},
    {"SSPRK-5-3", SSPRK_5_3},
    {"SSPRK-5-4", SSPRK_5_4},
};
static inline enum tableau_identifier str2tableau( std::string name)
{
    if( str2id.find(name) == str2id.end())
        throw dg::Error(dg::Message(_ping_)<<"Tableau "<<name<<" not found!");
    else
        return str2id[name];
}
static inline std::string tableau2str( enum tableau_identifier id)
{
    for( auto name: str2id)
    {
        if( name.second == id)
            return name.first;
    }
    throw dg::Error(dg::Message(_ping_)<<"Tableau conversion failed!");
}
 
template<class real_type>
ShuOsherTableau<real_type> shuosher_tableau( enum tableau_identifier id)
{
    switch(id)
    {
        case SSPRK_2_2:
            return dg::tableau::ssprk_2_2<real_type>();
        case SSPRK_3_2:
            return dg::tableau::ssprk_3_2<real_type>();
        case SSPRK_3_3:
            return dg::tableau::ssprk_3_3<real_type>();
        case SSPRK_5_3:
            return dg::tableau::ssprk_5_3<real_type>();
        case SSPRK_5_4:
            return dg::tableau::ssprk_5_4<real_type>();
        default:
            throw dg::Error(dg::Message(_ping_)<<"Tableau "<<tableau2str(id)<<" is not in Shu-Osher form!");
    }
    return ShuOsherTableau<real_type>(); //avoid compiler warning
}
template<class real_type>
ButcherTableau<real_type> tableau( enum tableau_identifier id)
{
    switch(id){
        case EXPLICIT_EULER_1_1:
            return dg::tableau::explicit_euler_1_1<real_type>();
        case MIDPOINT_2_2:
            return dg::tableau::midpoint_2_2<real_type>();
        case KUTTA_3_3:
            return dg::tableau::kutta_3_3<real_type>();
        case CLASSIC_4_4:
            return dg::tableau::classic_4_4<real_type>();
        case HEUN_EULER_2_1_2:
            return dg::tableau::heun_euler_2_1_2<real_type>();
        case CAVAGLIERI_3_1_2:
            return dg::tableau::cavaglieri_exp_3_1_2<real_type>();
        case FEHLBERG_3_2_3:
            return dg::tableau::fehlberg_3_2_3<real_type>();
        case FEHLBERG_4_2_3:
            return dg::tableau::fehlberg_4_2_3<real_type>();
        case BOGACKI_SHAMPINE_4_2_3:
            return dg::tableau::bogacki_shampine_4_2_3<real_type>();
        case CAVAGLIERI_4_2_3:
            return dg::tableau::cavaglieri_exp_4_2_3<real_type>();
        case ARK324L2SA_ERK_4_2_3:
            return dg::tableau::ark324l2sa_erk_4_2_3<real_type>();
        case ZONNEVELD_5_3_4:
            return dg::tableau::zonneveld_5_3_4<real_type>();
        case ARK436L2SA_ERK_6_3_4:
            return dg::tableau::ark436l2sa_erk_6_3_4<real_type>();
        case SAYFY_ABURUB_6_3_4:
            return dg::tableau::sayfy_aburub_6_3_4<real_type>();
        case CASH_KARP_6_4_5:
            return dg::tableau::cash_karp_6_4_5<real_type>();
        case FEHLBERG_6_4_5:
            return dg::tableau::fehlberg_6_4_5<real_type>();
        case DORMAND_PRINCE_7_4_5:
            return dg::tableau::dormand_prince_7_4_5<real_type>();
        case TSITOURAS09_7_4_5:
            return dg::tableau::tsitouras09_7_4_5<real_type>();
        case TSITOURAS11_7_4_5:
            return dg::tableau::tsitouras11_7_4_5<real_type>();
        case ARK548L2SA_ERK_8_4_5:
            return dg::tableau::ark548l2sa_erk_8_4_5<real_type>();
        case VERNER_9_5_6:
            return dg::tableau::verner_9_5_6<real_type>();
        case VERNER_10_6_7:
            return dg::tableau::verner_10_6_7<real_type>();
        case FEHLBERG_13_7_8:
            return dg::tableau::fehlberg_13_7_8<real_type>();
        case DORMAND_PRINCE_13_7_8:
            return dg::tableau::dormand_prince_13_7_8<real_type>();
        case FEAGIN_17_8_10:
            return dg::tableau::feagin_17_8_10<real_type>();
        case IMPLICIT_EULER_1_1:
            return dg::tableau::implicit_euler_1_1<real_type>();
        case IMPLICIT_MIDPOINT_1_2:
            return dg::tableau::implicit_midpoint_1_2<real_type>();
        case TRAPEZOIDAL_2_2:
            return dg::tableau::trapezoidal_2_2<real_type>();
        case SDIRK_2_1_2:
            return dg::tableau::sdirk_2_1_2<real_type>();
        case CAVAGLIERI_IMPLICIT_3_1_2:
            return dg::tableau::cavaglieri_imp_3_1_2<real_type>();
        case BILLINGTON_3_3_2:
            return dg::tableau::billington_3_3_2<real_type>();
        case TRBDF2_3_3_2:
            return dg::tableau::trbdf2_3_3_2<real_type>();
        case KVAERNO_4_2_3:
            return dg::tableau::kvaerno_4_2_3<real_type>();
        case SDIRK_4_2_3:
            return dg::tableau::sdirk_4_2_3<real_type>();
        case CAVAGLIERI_IMPLICIT_4_2_3:
            return dg::tableau::cavaglieri_imp_4_2_3<real_type>();
        case ARK324L2SA_DIRK_4_2_3:
            return dg::tableau::ark324l2sa_dirk_4_2_3<real_type>();
        case CASH_5_2_4:
            return dg::tableau::cash_5_2_4<real_type>();
        case CASH_5_3_4:
            return dg::tableau::cash_5_3_4<real_type>();
        case SDIRK_5_3_4:
            return dg::tableau::sdirk_5_3_4<real_type>();
        case KVAERNO_5_3_4:
            return dg::tableau::kvaerno_5_3_4<real_type>();
        case ARK436L2SA_DIRK_6_3_4:
            return dg::tableau::ark436l2sa_dirk_6_3_4<real_type>();
        case KVAERNO_7_4_5:
            return dg::tableau::kvaerno_7_4_5<real_type>();
        case ARK548L2SA_DIRK_8_4_5:
            return dg::tableau::ark548l2sa_dirk_8_4_5<real_type>();
        case SANCHEZ_3_3:
            return dg::tableau::sanchez_3_3<real_type>();
        case SANCHEZ_3_4:
            return dg::tableau::sanchez_3_4<real_type>();
        case SANCHEZ_6_5:
            return dg::tableau::sanchez_6_5<real_type>();
        case SANCHEZ_7_6:
            return dg::tableau::sanchez_7_6<real_type>();
        default:
            return ButcherTableau<real_type>(shuosher_tableau<real_type>(id));
    }
    return ButcherTableau<real_type>(); //avoid compiler warning
}
 
 
template<class real_type>
ShuOsherTableau<real_type> shuosher_tableau( std::string name)
{
        return shuosher_tableau<real_type>( str2tableau(name));
}
template<class real_type>
ButcherTableau<real_type> tableau( std::string name)
{
        return tableau<real_type>( str2tableau(name));
}
 
}//namespace create
 
template<class real_type>
struct ConvertsToButcherTableau
{
    using value_type = real_type;
    ConvertsToButcherTableau( ButcherTableau<real_type> tableau): m_t(tableau){}
 
    ConvertsToButcherTableau( enum tableau_identifier id):m_t( dg::create::tableau<real_type>(id)){}
    ConvertsToButcherTableau( std::string name):m_t( dg::create::tableau<real_type>(name)){}
    ConvertsToButcherTableau( const char* name):m_t( dg::create::tableau<real_type>(std::string(name))){}
    operator ButcherTableau<real_type>( )const{
        return m_t;
    }
    private:
    ButcherTableau<real_type> m_t;
};
 
template<class real_type>
struct ConvertsToShuOsherTableau
{
    using value_type = real_type;
    ConvertsToShuOsherTableau( ShuOsherTableau<real_type> tableau): m_t(tableau){}
 
    ConvertsToShuOsherTableau( enum tableau_identifier id):m_t( dg::create::shuosher_tableau<real_type>(id)){}
    ConvertsToShuOsherTableau( std::string name):m_t( dg::create::shuosher_tableau<real_type>(name)){}
    ConvertsToShuOsherTableau( const char* name):m_t( dg::create::shuosher_tableau<real_type>(std::string(name))){}
    operator ShuOsherTableau<real_type>( )const{
        return m_t;
    }
    private:
    ShuOsherTableau<real_type> m_t;
};
 
}//namespace dg