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| | MultistepTableau () |
| | No memory allocation. More...
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| | MultistepTableau (unsigned steps, unsigned order, const std::vector< real_type > &a_v, const std::vector< real_type > &b_v, const std::vector< real_type > &c_v) |
| | Construct a tableau. More...
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| real_type | a (unsigned i) |
| | Read the a_i coefficients. More...
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| real_type | ex (unsigned i) |
| | Read the explicit (b_i) coefficients. More...
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| real_type | im (unsigned i) |
| | Read the implicit (c_i) coefficients. More...
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| unsigned | steps () const |
| | The number of stages s. More...
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| unsigned | order () const |
| | global order of accuracy for the method More...
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| bool | isExplicit () const |
| | True if any of the explicit coefficients b_i are non-zero. More...
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| bool | isImplicit () const |
| | True if any of the implicit coefficients c_i are non-zero. More...
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template<class real_type>
struct dg::MultistepTableau< real_type >
Manage coefficients of Multistep methods.
The general s-step multistep method has the form
\[ y^{n+1} = \sum_{i=0}^{s-1} a_i y^{n-i} + h \sum_{i=0}^{s-1} b_i E( t_{n-i}, y_{n-i}) + h \sum_{i=0}^s c_i I( t_{n+1-i}, y^{n+1-i}) \]
where E is the explicit and I is implicit part. A purely implicit method is one where all \( b_i\) are zero, while an explicit one is one where all \( c_i\) are zero. A tableau thus consists of the three arrays a, b and c the number of steps and the order of the method. Currently available methods are:
ImEx methods
We follow the naming convention as NAME-S-Q
- NAME is the author or name of the method
- S is the number of steps in the method
- Q is the global order of the method
| Name | Identifier | Description |
| ImEx-Euler-1-1 | dg::IMEX_EULER_1_1 | Explicit Euler combined with Implicit Euler |
| Euler | dg::IMEX_EULER_1_1 | For convenience |
| ImEx-Koto-2-2 | dg::IMEX_KOTO_2_2 | Koto T. Front. Math. China 2009, 4(1): 113-129 A stabilized 2nd order scheme with a large region of stability |
| ImEx-Adams-X-X | dg::IMEX_ADAMS_X_X | Hundsdorfer and Ruuth, Journal of Computational Physics 225 (2007) - Note
- Possible values for X: 2 (C=0.44), 3 (C=0.16)
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| ImEx-BDF-X-X | dg::IMEX_BDF_X_X | The family of schems described in Hundsdorfer and Ruuth, Journal of Computational Physics 225 (2007)
The implicit part is a normal BDF scheme https://en.wikipedia.org/wiki/Backward_differentiation_formula while the explicit part equals the Minimal Projecting method by Alfeld, P., Math. Comput. 33.148 1195-1212 (1979) or extrapolated BDF in Hundsdorfer, W., Ruuth, S. J., & Spiteri, R. J. (2003). Monotonicity-preserving linear multistep methods. SIAM Journal on Numerical Analysis, 41(2), 605-623
- Note
- Possible values for X: 1 (C=1.00), 2 (C=0.63), 3 (C=0.39), 4 (C=0.22), 5 (C=0.09), 6
Note that X=3 is identical to the "Karniadakis" scheme
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| Karniadakis | dg::IMEX_BDF_3_3 | The ImEx-BDF-3-3 scheme is identical to the widely used "Karniadakis" scheme Karniadakis, et al. J. Comput. Phys. 97 (1991) |
| ImEx-TVB-X-X | dg::IMEX_TVB_X_X | The family of schems described in < Hundsdorfer and Ruuth, Journal of Computational Physics 225 (2007)
The explicit part is a TVB scheme while the implicit part is optimized to maximize damping of high wavelength
- Note
- Possible values for X: 3 (C=0.54), 4 (C=0.46), 5 (C=0.38)
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- Note
- the CFL coefficient C is given relative to the forward Euler method: \( \Delta t < C \Delta t_{FE}\).
- Attention
- The coefficient C is the one that ensures the TVD property of the scheme and is not directly related to the stability region of the scheme
- Note
- ImEx multistep tableaus can be used in ExplicitMultistep, ImplicitMultistep and ImExMultistep
Explicit methods
We follow the naming convention as NAME-S-Q
- NAME is the author or name of the method
- S is the number of steps in the method
- Q is the global order of the method
| Name | Identifier | Description |
| AB-X-X | dg::AB_X_X | The family of schemes described in Linear multistep methods as Adams-Bashforth
\[ u^{n+1} = u^n + \Delta t\sum_{j=0}^{s-1} b_j f\left(t^n - j \Delta t, u^{n-j}\right) \]
- Note
- Possible stages are X: 1, 2,..., 5, the order of the method is the same as its stages
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The Adams-Bashforth schemes implemented here need less storage but may have a smaller region of absolute stability than for example an extrapolated BDF method of the same order.
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| eBDF-X-X | dg::eBDF_X_X | The family of schemes described in Hundsdorfer, W., Ruuth, S. J., & Spiteri, R. J. (2003). Monotonicity-preserving linear multistep methods. SIAM Journal on Numerical Analysis, 41(2), 605-623 as extrapolated BDF where it is found to be TVB (total variation bound). The schemes also appear as Minimal Projecting scheme described in Alfeld, P., Math. Comput. 33.148 1195-1212 (1979)
- Note
- Possible stages are X: 1 (C=1), 2 (C=0.63), 3 (C=0.39), 4 (C=0.22), 5 (C=0.09), 6 with the order the same as the number of stages
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| TVB-X-X | dg::TVB_X_X | The family of schemes described in S.J. Ruuth and W. Hundsdorfer, High-order linear multistep methods with general monotonicity and boundedness properties, Journal of Computational Physics, Volume 209, Issue 1, 2005 as Total variation Bound. These schemes have larger allowable step sizes than the eBDF family,
- Note
- Possible values for X are 1 (C=1), 2 (C=0.5), 3 (C=0.54), 4 (C=0.46), 5 (C=0.38) 6 (C=0.33). We highlight that TVB-3-3 has 38% larger allowable stepsize than eBDF-3-3 and TVB-4-4 has 109% larger stepsize than eBDF-4-4 (to ensure the TVB property, not stability).
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| SSP-X-Y | dg::SSP_X_Y | The family of schemes described in Gottlieb, S. On high order strong stability preserving runge-kutta and multi step time discretizations. J Sci Comput 25, 105–128 (2005) as Strong Stability preserving. We implement the lowest order schemes for each stage and disregard the remaining schemes in the paper since their CFL conditions are worse than the TVB scheme of the same order. - Note
- Possible values for X-Y : 1-1 (C=1), 2-2 (C=0.5), 3-2 (C=0.5), 4-2 (C=0.66), 5-3 (C=0.5), 6-3 (C=0.567).
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These schemes are noteworthy because the coefficients b_i are all positive except for the 2-2 method and the "4-2" and "6-3" methods allow slightly larger allowable stepsize but increased storage requirements than TVB of same order (2 and 3).
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- Note
- Total variation bound (TVB) means \( || v^n|| \leq M ||v^0||\) where the norm signifies the total variation semi-norm. Total variation diminishing (TVD) means M=1, and strong stability preserving (SSP) is the same as TVD, TVB schemes converge to the correct entropy solutions of hyperbolic conservation laws
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the CFL coefficient C is given relative to the forward Euler method: \( \Delta t < C \Delta t_{FE}\).
- Attention
- The coefficient C is the one that ensures the TVD property of the scheme and is not directly related to the stability region of the scheme
- Template Parameters
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| real_type | type of the coefficients |
- See also
- ExplicitMultistep, ImplicitMultistep, ImExMultistep
1.9.3