Strong stability for additive Runge-Kutta methods

SIAM Journal on Numerical Analysis

Volumn 44, Issue 4, 2006, Pages 1735-1758

  Higueras, Inmaculada ^a  

^a PUBLIC UNIVERSITY OF NAVARRE   (Spain)

Author keywords

Absolutely monotonic; CFL coefficient; Radius of absolute monotonicity; Runge Kutta; Strong stability preserving

Indexed keywords

NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; RUNGE KUTTA METHODS;

ABSOLUTE MONOTONICITY; CFL COEFFICIENTS; STRONG STABILITY PRESERVING;

SYSTEM STABILITY;

EID: 34547535337     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/040612968     Document Type: Article

References (27)

1. A. L ARAÚJO, A. MURUA, AND J. M. SANZ-SERNA, Symplectic methods based on decompositions, SIAM J. Numer. Anal., 34 (1997), pp. 1926-1947.
2. U. M. ASCHER, S. J. RUUTH, AND R. SPITERI, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25 (1997), pp. 151-167.
3. R. E. CAFLISCH, S. JIN, AND G. RUSSO, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal., 34 (1997), pp. 246-281.
4. M. P. CALVO, J. DE FRUTOS, AND J. NOVO, Linearly implicit Runge-Kutta methods for advection-reaction- diffusion equations, Appl. Numer. Math., 37 (2001), pp. 535-549.
5. L. FERRACINA AND M. N. SPIJKER, Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods, SIAM J. Numer. Anal., 42 (2004), pp. 1073-1093.
6. L. FERRACINA AND M. N. SPIJKER, An extension and analysis of the Shu-Osher representation of Runge-Kutta methods, Math. Comp., 74 (2005), pp. 201-219.
7. A. GERISCH, D. F. GRIFFITHS, R. WEINER, AND M. A. J. CHAPLAIN, A positive splitting method for mixed hyperbolic-parabolic systems, Numer. Methods Partial Differential Equations, 17 (2001), pp. 152-168.
8. A. GERISCH AND R. WEINER, On the positivity of low order explicit Runge-Kutta schemes in splitting methods, Comput. Math. Appl., 45 (2003), pp. 53-67.
9. S. GOTTLIEB, C.-W. SHU, AND E. TADMOR, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89-112.
10. S. GOTTLIEB AND C. W. SHU, Total variation diminishing Runge-Kutta schemes, Math. Comp., 67 (1998), pp. 73-85.
11. E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integration, Springer Ser. Comput. Math. 31, Springer-Verlag, Berlin, 2003.
12. I. HIGUERAS, On strong stability preserving time discretization methods, J. Sei. Comput., 21 (2004), pp. 193-223.
13. I. HIGUERAS, Representations of Runge-Kutta methods and strong stability preserving methods, SIAM J. Numer. Anal., 43 (2005), pp. 924-948.
14. I. HIGUERAS AND T. ROLDAN, On strong stability for additive Runge-Kutta methods: Entropy monotonicity for the Broadwell model, submitted.
15. W. HUNDSDORFER AND J. G. VERWER, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Ser. Comput. Math. 33, Springer-Verlag, Berlin, 2003.
16. P. LANCASTER AND M. TISMENETSKY, The Theory of Matrices, Academic Press, San Diego, CA, 1985.
17. C. A. KENNEDY AND M. H. CARPENTER, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), pp. 139-181.
18. J. F. B. M. KRAAIJEVANGER, Contractivity of Runge-Kutta methods, BIT, 31 (1991), pp. 482-528.
19. L. PARESCHI AND G. RUSSO, Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations, in Recent Trends in Numerical Analysis, Adv. Theory Comput. Math. 3, L. Brugnano and D. Trigiante, eds., Nova Sci. Publ., Huntington, NY, 2001, pp. 269-288.
20. L. PARESCHI AND G. RUSSO, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., to appear.
21. S. J. RUUTH AND R. J. SPITERI, Two barriers on strong stability preserving time discretization methods, J. Sei. Comput., 17 (2002), pp. 211-220.
22. J. W. SHEN AND X. ZHONG, Semi-Implicit Runge-Kutta Schemes for Non-Autonomous Deferential Equations in Reactive Flow Computations, Paper 1996-1969, 27th AIAA Fluid Dynamics Conference, New Orleans, LA, June 17-20, 1996.
23. C. W. SHU AND S. OSHER, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439-471.
24. C.-W. SHU, Total-variation-diminishing time discretizations, SIAM J. Sci. Comput., 9 (1988), pp. 1073-1084.
25. C.-W. SHU, A survey of strong stability preserving high order time discretizations, in Collected Lectures on the Preservation of Stability under Discretization, D. Estep and S. Tavener, eds., Proc. Appl. Math. 109, SIAM, Philadelphia, 2002, pp. 51-65.
26. R. J. SPITERI AND S. J. RUUTH, A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40 (2002), pp. 469-491.
27. X. ZHONG, Additive semi-implicit Runge-Kutta methods for computing high speed nonequilibrium reactive flows, J. Comput. Phys., 128 (1996), pp. 19-31.