Strong Stability Preserving IMEX Methods for Partitioned Systems of Diferential Equations

doi:10.1007/s42967-021-00158-x

Abstract

Abstract: We investigate strong stability preserving (SSP) implicit-explicit (IMEX) methods for partitioned systems of diferential equations with stif and nonstif subsystems. Conditions for order p and stage order q = p are derived, and characterization of SSP IMEX methods is provided following the recent work by Spijker. Stability properties of these methods with respect to the decoupled linear system with a complex parameter, and a coupled linear system with real parameters are also investigated. Examples of methods up to the order p = 4 and stage order q = p are provided. Numerical examples on six partitioned test systems confrm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration, and they are also suitable for preserving the accuracy in the stif limit or preserving the positivity of the numerical solution for large stepsizes.

Key words: Partitioned systems of diferential equations, SSP property, IMEX general linear methods, Construction of highly stable methods

CLC Number:

65L05

Giuseppe Izzo, Zdzisław Jackiewicz. Strong Stability Preserving IMEX Methods for Partitioned Systems of Diferential Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 719-758.

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References

1. Asher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial diferential equations. Appl. Numer. Math. 25, 151–167 (1997)
2. Asher, U.M., Ruuth, S.J., Wetton, B.: Implicit-explicit methods for time dependent PDE’s. SIAM J. Numer. Anal. 32, 797–823 (1995)
3. Butcher, J.C.: Diagonally-implicit multi-stage integration methods. Appl. Numer. Math. 11, 347–363 (1993)
4. Butcher, J.C., Jackiewicz, Z.: Diagonally implicit general linear methods for ordinary diferential equations. BIT 33, 452–472 (1993)
5. Butcher, J.C., Wright, W.M.: The construction of practical general linear methods. BIT 43, 695–721 (2003)
6. Califano, G., Izzo, G., Jackiewicz, Z.: Starting procedures for general linear methods. Appl. Numer. Math. 120, 165–175 (2017)
7. Califano, G., Izzo, G., Jackiewicz, Z.: Strong stability preserving general linear methods with RungeKutta stability. J. Sci. Comput. 76, 943–968 (2018)
8. Cardone, A., Jackiewicz, Z., Sandu, A., Zhang, H.: Extrapolated implicit-explicit Runge-Kutta methods. Math. Model. Anal. 19, 18–43 (2014)
9. Conde, S., Gottlieb, S., Grant, Z.J., Shadid, J.N.: Implicit and implicit-explicit strong stability preserving Runge-Kutta methods with high linear order. J. Sci. Comput. 73, 667–690 (2017)
10. Constantinescu, E.M., Sandu, A.: Optimal strong-stability-preserving general linear methods. SIAM J. Sci. Comput. 32, 3130–3150 (2010)
11. Enright, W.H.: Second derivative multistep methods for stif ordinary diferential equations. SIAM J. Numer. Anal. 11, 321–331 (1974)
12. Ferracina, L., Spijker, M.N.: An extension and analysis of the Shu-Osher representation of RungeKutta methods. Math. Comput. 74, 201–219 (2004)
13. Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods. SIAM J. Numer. Anal. 42, 1073–1093 (2004)
14. Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-boundedness in general RungeKutta procedures. Appl. Numer. Math. 53, 265–279 (2005)
15. Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge-Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)
16. Gottlieb, S.: On high order strong stability preserving Runge-Kutta methods and multistep time discretizations. J. Sci. Comput. 25, 105–127 (2005)
17. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38, 251–289 (2009)
18. Gottlieb, S., Ketcheson, D., Shu, C.-W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientifc, New Jersey (2011)
19. Gottlieb, S., Ruuth, S.J.: Optimal strong-stability-preserving time stepping schemes with fast downwind spatial discretizations. J. Sci. Comput. 27, 289–303 (2006)
20. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
21. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Diferential Equations I: Nonstif Problems. Springer-Verlag, New York (1993)
22. Hairer, E., Wanner, G.: Solving Ordinary Diferential Equations Ⅱ. Stif and Diferential-Algebraic Problems. Springer Verlag, Berlin (1996)
23. Higueras, I.: On strong stability preserving time discretization methods. J. Sci. Comput. 21, 193–223 (2004)
24. Higueras, I.: Monotonicity for Runge-Kutta methods: inner product norms. J. Sci. Comput. 24, 97–117 (2005)
25. Higueras, I.: Representations of Runge-Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43, 924–948 (2005)
26. Higueras, I., Happenhofer, N., Koch, O., Kupka, F.: Optimized strong stability preserving IMEX Runge-Kutta methods. J. Comput. Appl. Math. 272, 116–140 (2014)
27. Hofer, E.: A partially implicit method for large stif systems of ODE’s with only few equations introducing small time-constants. SIAM J. Numer. Anal. 13, 645–663 (1976)
28. Hundsdorfer, W., Ruuth, S.J.: On monotonicity and boundedness properties of linear multistep methods. Math. Comput. 75, 655–672 (2005)
29. Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225, 2016–2042 (2007)
30. Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605–623 (2003)
31. Izzo, G., Jackiewicz, Z.: Strong stability preserving general linear methods. J. Sci. Comput. 65, 271– 298 (2015)
32. Izzo, G., Jackiewicz, Z.: Strong stability preserving multistage integration methods. Math. Model. Anal. 20, 552–577 (2015)
33. Izzo, G., Jackiewicz, Z.: Highly stable implicit-explicit Runge-Kutta methods. Appl. Numer. Math. 113, 71–92 (2017)
34. Izzo, G., Jackiewicz, Z.: Strong stability preserving transformed DIMSIMs. J. Comput. Appl. Math. 343, 174–188 (2018)
35. Izzo, G., Jackiewicz, Z.: Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part. Numer. Algorithms 81, 1343–1359 (2019)
36. Jackiewicz, Z.: General Linear Methods for Ordinary Diferential Equations. John Wiley, Hoboken (2009)
37. Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge-Kutta methods for ordinary diferential equations. SIAM J. Numer. Anal. 32, 1390–1427 (1995)
38. Jin, S.: Runge-Kutta methods for hyperbolic systems with stif relaxation terms. J. Comput. Phys. 122, 51–67 (1995)
39. Ketcheson, D.I., Gottlieb, S., Macdonald, C.B.: Strong stability preserving two-step Runge-Kutta methods. SIAM J. Numer. Anal. 49, 2618–2639 (2011)
40. Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)
41. Ruuth, S.J., Hundsdorfer, W.: High-order linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 209, 226–248 (2005)
42. Shu, C.-W.: High order ENO and WENO schemes for computational fuid dynamics. In: Barth, T.J., Deconinck, H. (eds) High-Order Methods for Computational Physics. Lecture Notes in Computational Science and Engineering, vol. 9, pp. 439–582. Springer, Berlin (1999)
43. Shu, C.-W., Osher, S.: Efcient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
44. Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45, 1226–1245 (2007)
45. Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)
46. Van der Houwen, P.J.: Explicit Runge-Kutta formulas with increased stability boundaries. Numer. Math. 20, 149–164 (1972)
47. Wright, W.: General linear methods with inherent Runge-Kutta stability. Ph.D. thesis. The University of Auckland, New Zealand (2002)
48. Wright, W.: Explicit general linear methods with inherent Runge-Kutta stability. Numer. Algorithms 31, 381–399 (2002)
49. Zhang, H., Sandu, A., Blaise, S.: Partitioned and implicit-explicit general linear methods for ordinary diferential equations. J. Sci. Comput. 61, 119–144 (2014)