Optimal implicit strong stability preserving Runge-Kutta methods

Applied Numerical Mathematics

Volumn 59, Issue 2, 2009, Pages 373-392

  Ketcheson, David I ^a   Macdonald, Colin B ^b   Gottlieb, Sigal ^c  

^a UNIVERSITY OF WASHINGTON   (United States)

^b SIMON FRASER UNIVERSITY   (Canada)

^c UNIVERSITY OF MASSACHUSETTS DARTMOUTH   (United States)

Author keywords

Runge Kutta methods; Strong stability preserving; Time discretization

Indexed keywords

EULER EQUATIONS; NON NEWTONIAN FLOW; NUMERICAL METHODS; SETTLING TANKS;

BOUNDEDNESS; BOUNDEDNESS PROPERTIES; CONTRACTIVITY; DISCRETIZATIONS; EULER TIME INTEGRATIONS; EXPLICIT METHODS; NUMERICAL TESTS; OPTIMAL METHODS; SPATIAL DISCRETIZATIONS; STRONG STABILITIES; STRONG STABILITY PRESERVING; TIME DISCRETIZATION; TIME STEPS;

RUNGE KUTTA METHODS;

EID: 56249142763     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2008.03.034     Document Type: Article

References (34)

1. Butcher J.C. On the implementation of implicit Runge-Kutta methods. BIT 17 (1976) 375-378
2. G. Dahlquist, R. Jeltsch, Generalized disks of contractivity for explicit and implicit Runge-Kutta methods, Tech. rep., Dept. of Numer. Anal. and Comp. Sci., Royal Inst. of Techn., Stockholm, 1979
3. Dekker K., and Verwer J.G. Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs vol. 2 (1984), North-Holland Publishing Co., Amsterdam
4. Ferracina L., and Spijker M.N. Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods. SIAM Journal of Numerical Analysis 42 (2004) 1073-1093
5. L. Ferracina, M.N. Spijker, Computing optimal monotonicity-preserving Runge-Kutta methods, Tech. Rep. MI2005-07, Mathematical Institute, Leiden University, 2005
6. Ferracina L., and Spijker M.N. An extension and analysis of the Shu-Osher representation of Runge-Kutta methods. Mathematics of Computation 249 (2005) 201-219
7. Ferracina L., and Spijker M.N. Strong stability of singly-diagonally-implicit Runge-Kutta methods. Applied Numerical Mathematics 58 (2007) 1675-1686
8. Gottlieb S. On high order strong stability preserving Runge-Kutta and multi step time discretizations. Journal of Scientific Computing 25 (2005) 105-127
9. Gottlieb S., and Shu C.-W. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation 67 (1998) 73-85
10. Gottlieb S., Shu C.-W., and Tadmor E. Strong stability preserving high-order time discretization methods. SIAM Review 43 (2001) 89-112
11. Hairer E., Nørsett S.P., and Wanner G. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. second ed. vol. 8 (1993), Springer-Verlag, Berlin
12. Hairer E., and Wanner G. Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics. second ed. vol. 14 (1996), Springer-Verlag, Berlin
13. Higueras I. On strong stability preserving time discretization methods. Journal of Scientific Computing 21 (2004) 193-223
14. Higueras I. Representations of Runge-Kutta methods and strong stability preserving methods. SIAM Journal of Numerical Analysis 43 (2005) 924-948
15. Hundsdorfer W., and Verwer J. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics vol. 14 (2003), Springer
16. D.I. Ketcheson, An algebraic characterization of strong stability preserving Runge-Kutta schemes, B.Sc. thesis, Brigham Young University, Provo, Utah, USA, 2004
17. D.I. Ketcheson, Highly efficient strong stability preserving Runge-Kutta methods with low-storage implementations, SIAM Journal on Scientific Computing (2008), doi: 10.1137/07070485X
18. D.I. Ketcheson, High order numerical methods for wave propagation, unpublished doctoral thesis, University of Washington
19. Ketcheson D.I., Macdonald C.B., and Gottlieb S. Numerically optimal SSP Runge-Kutta methods. website (2007). http://www.cfm.brown.edu/people/sg/ssp.html
20. Ketcheson D.I., and Robinson A.C. On the practical importance of the SSP property for Runge-Kutta time integrators for some common Godunov-type schemes. International Journal for Numerical Methods in Fluids 48 (2005) 271-303
21. Kraaijevanger J.F.B.M. Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems. Numerische Mathematik 48 (1986) 303-322
22. Kraaijevanger J.F.B.M. Contractivity of Runge-Kutta methods. BIT 31 (1991) 482-528
23. LeVeque R.J. Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press
24. C.B. Macdonald, Constructing high-order Runge-Kutta methods with embedded strong-stability-preserving pairs, Master's thesis, Simon Fraser University, August 2003
25. Macdonald C.B., Gottlieb S., and Ruuth S.J. A numerical study of diagonally split Runge-Kutta methods for PDEs with discontinuities. Journal of Scientific Computing (2008) 10.1007/s10915-007-9180-6
26. Ruuth S.J. Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Mathematics of Computation 75 253 (2006) 183-207 (electronic)
27. Sahinidis N.V., and Tawarmalani M. BARON 7.2: Global Optimization of Mixed-Integer Nonlinear Programs. User's Manual, available at (2004). http://www.gams.com/dd/docs/solvers/baron.pdf
28. Shu C.-W. Total-variation diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing 9 (1988) 1073-1084
29. Shu C.-W. A survey of strong stability-preserving high-order time discretization methods. Collected Lectures on the Preservation of Stability under Discretization (2002), SIAM, Philadelphia, PA
30. Shu C.-W., and Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 77 (1988) 439-471
31. Spijker M.N. Contractivity in the numerical solution of initial value problems. Numerische Mathematik 42 (1983) 271-290
32. Spiteri R.J., and Ruuth S.J. A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM Journal of Numerical Analysis 40 (2002) 469-491
33. Spiteri R.J., and Ruuth S.J. Nonlinear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods. Mathematics and Computers in Simulation 62 (2003) 125-135
34. van de Griend J.A., and Kraaijevanger J.F.B.M. Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems. Numerische Mathematik 49 (1986) 413-424