Highly efficient strong stability-preserving Runge-Kutta methods with low-storage implementations

SIAM Journal on Scientific Computing

Volumn 30, Issue 4, 2007, Pages 2113-2136

  Ketcheson, David I ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Low storage; Method of lines; Monotonicity; Runge Kutta methods; Strong stability preserving

Indexed keywords

NONLINEAR EQUATIONS; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; STABILITY;

LOW STORAGE IMPLEMENTATIONS; LOW-STORAGE; METHOD OF LINES; MONOTONICITY; SEMIDISCRETIZATION; STABILITY PROPERTIES; STRONG STABILITY PRESERVING; STRONG STABILITY PRESERVING RUNGE-KUTTA METHODS; TIME STEP; TIME-INTEGRATION;

RUNGE KUTTA METHODS;

EID: 55549084557     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/07070485X     Document Type: Article

References (33)

1. J. C. BUTCHER, Numerical Methods for Ordinary Differential Equations, Wiley, New York, 2003.
2. G. DAHLQUIST AND R. JELTSCH, Generalized Disks of Contractivity for Explicit and Implicit Runge-Kutta Methods, Technical report, Department of Numerical Analysis and Computer Science, Royal Institute of Technology, Stockholm, 1979.
3. L. FERRACINA AND M. N. SPIJKER, Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods, S1AM J. Numer. Anal., 42 (2004), pp. 1073-1093.
4. L. FERRACINA AND M. N. SPIJKER, Computing Optimal Monotonicity-Preserving Runge-Kutta Methods, Technical report MI2005-07, Mathematical Institute, Leiden University, 2005.
5. L. FERRACINA AND M. N. SPIJKER, An extension and analysis of the Shu-Osher representation of Runge-Kutta methods, Math. Comp., 249 (2005), pp. 201-219.
6. S. GOTTLIEB, On high order strong stability preserving Runge-Kutta and multi step time discretizations, J. Sci. Comput., 25 (2005), pp. 105-127.
7. S. GOTTLIEB AND L.-AD J. GOTTLIEB, Strong stability preserving properties of Rwige-Kutta time discretization methods for linear constant coefficient operators, J. Sci. Comput, 18 (2003), pp. 83-109.
8. S. GOTTLIEB AND C.-W. SHU, Total variation diminishing Runge-Kutta schemes, Math. Comp., 67 (1998), pp. 73-85.
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. E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Ser. Comput. Math. 14, Springer-Verlag, Berlin, 1991.
11. I. HIGUERAS, On strong stability preserving time discretization methods, J. Sci. Comput., 21 (2004), pp. 193-223.
12. I. HLGUERAS, Monotonicity for Runge-Kutta methods: Inner product norms, J. Sci. Comput., 24 (2005), pp. 97-117.'
13. I. HIGUERAS, Representations of Runge-Kutta methods and strong stability preserving methods, SIAM J. Numer. Anal., 43 (2005), pp. 924-948.
14. W. HUNDSDORFER, S. J. RUUTH, AND R. J. SPITERI, Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal., 41 (2003), pp. 605-623.
15. C. A. KENNEDY, M. H. CARPENTER, AND R. M. LEWIS, Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations, Appl. Numer. Math., 35 (2000), pp. 177-219.
16. D. I. KETCHESON, C. B. MACDONALD, AND S. GOTTLIEB, Optimal implicit strong stability preserving Runge-Kutta methods, submitted.
17. D. I. KETCHESON AND A. C. ROBINSON, On the practical importance of the SSP property for Runge-Kutta time integrators for some common Godunov-type schemes, Internat. J. Numer. Methods Fluids, 48 (2005), pp. 271-303.
18. J. F. B. M. KRAAIJEVANGER, Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems, Numer. Math., 48 (1986), pp. 303-322.
19. J. F. B. M. KRAAIJEVANGER, Contractimty of Runge-Kutta methods, BIT, 31 (1991), pp. 482-528.
20. E. J. KUBATKO, J. J. WESTERINK, AND C. DAWSON, Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge-Kutta time discretizations, J. Comput. Phys., 222 (2007), pp. 832-848.
21. C. B. MACDONALD, Constructing High-Order Runge-Kutta Methods with Embedded Strong-Stability-Preserving Pairs, Master's thesis, Simon Fraser University, Burnaby, BC, 2003.
22. S. RUUTH, Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Math. Comp., 75 (2006), pp. 183-207.
23. S. J. RUUTH AND W. HUNDSDORFER, High-order linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 209 (2005), pp. 226-248.
24. S. J. RUUTH AND R. J. SPITERI, Two barriers on strong-stability-preserving time discretization methods, J. Sci. Comput., 17 (2002), pp. 211-220.
25. S. J. RUUTH AND R. J. SPITERI, High-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations, SIAM J. Numer. Anal., 42 (2004), pp. 974-996.
26. C.-W. SHU, Total-variation-diminishing time discretizations, SIAM J. Sci. Comput., 9 (1988), pp. 1073-1084.
27. C.-W. SHU AND S. OSHER., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Conrput. Phys., 77 (1988), pp. 439-471.
28. M. N. SPIJKER, Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), pp. 271-290.
29. 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.
30. R. J. SPITERI AND S. J. RUUTH, Nonlinear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods, Math. Comput. Simulation, 62 (2003), pp. 125-135.
31. J. G. VERWER., Explicit Runge-Kutta methods for parabolic partial differential equations, Appl. Numer. Math., 22 (1996), pp. 359-379.
32. R. WANG AND R. J. SPITERI, Linear instability of the fifth-order WENO method, SIAM J. Numer. Anal., 45 (2007), pp. 1871-1901.
33. J. H. WILLIAMSON, Low-storage Runge-Kutta schemes, J. Comput. Phys., 35 (1980), pp. 48-56.