Strong-stability-preserving additive linear multistep methods

Hadjimichael, Yiannis; Ketcheson, David

doi:10.1090/mcom/3296

Strong-stability-preserving additive linear multistep methods
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by Yiannis Hadjimichael and David I. Ketcheson;

Math. Comp. 87 (2018), 2295-2320

DOI: https://doi.org/10.1090/mcom/3296

Published electronically: February 20, 2018

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Abstract:

The analysis of strong-stability-preserving (SSP) linear multistep methods is extended to semi-discretized problems for which different terms on the right-hand side satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems. Optimal perturbed methods attain larger monotonicity-preserving step sizes when the different forward Euler conditions are taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding nonadditive SSP linear multistep methods.

References

R. Donat, I. Higueras, and A. Martínez-Gavara, On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms, Math. Comp. 80 (2011), no. 276, 2097–2126. MR 2813350, DOI 10.1090/S0025-5718-2011-02463-4
Thor Gjesdal, Implicit-explicit methods based on strong stability preserving multistep time discretizations, Appl. Numer. Math. 57 (2007), no. 8, 911–919. MR 2331084, DOI 10.1016/j.apnum.2006.09.001
Sigal Gottlieb, David Ketcheson, and Chi-Wang Shu, Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MR 2789749, DOI 10.1142/7498
Sigal Gottlieb and Steven J. Ruuth, Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations, J. Sci. Comput. 27 (2006), no. 1-3, 289–303. MR 2285782, DOI 10.1007/s10915-005-9054-8
Nicholas J. Higham, Accuracy and stability of numerical algorithms, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR 1927606, DOI 10.1137/1.9780898718027
Inmaculada Higueras, Strong stability for additive Runge-Kutta methods, SIAM J. Numer. Anal. 44 (2006), no. 4, 1735–1758. MR 2257125, DOI 10.1137/040612968
I. Higueras, Positivity properties for the classical fourth order Runge-Kutta method, Proceedings of the MCalv6(5) Conference, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, vol. 33, Acad. Cienc. Exact. Fís. Quím. Nat. Zaragoza, Zaragoza, 2010, pp. 125–139. MR 3346421
Willem Hundsdorfer and Steven J. Ruuth, IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys. 225 (2007), no. 2, 2016–2042. MR 2349693, DOI 10.1016/j.jcp.2007.03.003
Willem Hundsdorfer, Steven J. Ruuth, and Raymond J. Spiteri, Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal. 41 (2003), no. 2, 605–623. MR 2004190, DOI 10.1137/S0036142902406326
David I. Ketcheson, Computation of optimal monotonicity preserving general linear methods, Math. Comp. 78 (2009), no. 267, 1497–1513. MR 2501060, DOI 10.1090/S0025-5718-09-02209-1
David I. Ketcheson, Step sizes for strong stability preservation with downwind-biased operators, SIAM J. Numer. Anal. 49 (2011), no. 4, 1649–1660. MR 2831065, DOI 10.1137/100818674
H. W. J. Lenferink, Contractivity preserving explicit linear multistep methods, Numer. Math. 55 (1989), no. 2, 213–223. MR 987386, DOI 10.1007/BF01406515
H. W. J. Lenferink, Contractivity-preserving implicit linear multistep methods, Math. Comp. 56 (1991), no. 193, 177–199. MR 1052098, DOI 10.1090/S0025-5718-1991-1052098-0
R. J. LeVeque and H. C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys. 86 (1990), no. 1, 187–210. MR 1033905, DOI 10.1016/0021-9991(90)90097-K
Robert H. Martin Jr., Nonlinear operators and differential equations in Banach spaces, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 492671
Olavi Nevanlinna and Werner Liniger, Contractive methods for stiff differential equations. I, BIT 18 (1978), no. 4, 457–474. MR 520755, DOI 10.1007/BF01932025
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1996.
Steven J. Ruuth and Willem Hundsdorfer, High-order linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys. 209 (2005), no. 1, 226–248. MR 2145787, DOI 10.1016/j.jcp.2005.02.029
Jørgen Sand, Circle contractive linear multistep methods, BIT 26 (1986), no. 1, 114–122. MR 833836, DOI 10.1007/BF01939367
Chi-Wang Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 6, 1073–1084. MR 963855, DOI 10.1137/0909073
Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR 954915, DOI 10.1016/0021-9991(88)90177-5
M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math. 42 (1983), no. 3, 271–290. MR 723625, DOI 10.1007/BF01389573