An extension and analysis of the Shu-Osher representation of Runge-Kutta methods

In the context of solving nonlinear partial differential equations, Shu and Osher introduced representations of explicit Runge-Kutta methods, which lead to stepsize conditions under which the numerical process is total-variation-diminishing (TVD). Much attention has been paid to these representations in the literature. In general, a Shu-Osher representation of a given Runge-Kutta method is not unique. Therefore, of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them. Several basic questions are still open, notably regarding the following issues: (1) the formulation of a simple and general strategy for finding a best possible Shu-Osher representation for any given Runge-Kutta method; (2) the question of whether the TVD property of a given Runge-Kutta method can still be guaranteed when the stepsize condition, corresponding to a best possible Shu-Osher representation of the method, is violated; (3) the generalization of the Shu-Osher approach to general (possibly implicit) Runge-Kutta methods. In this paper we give an extension and analysis of the original Shu-Osher representation, by means of which the above questions can be settled. Moreover, we clarify analogous questions regarding properties which are referred to, in the literature, by the terms monotonicity and strong-stability-preserving (SSP).

[1]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

[2]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[3]  M. N. Spijker,et al.  Stepsize Restrictions for the Total-Variation-Diminishing Property in General Runge-Kutta Methods , 2004, SIAM J. Numer. Anal..

[4]  K. W. Morton.  Stability of finite difference approximations to a diffusion–convection equation , 1980 .

[5]  Inmaculada Higueras,et al.  On Strong Stability Preserving Time Discretization Methods , 2004, J. Sci. Comput..

[6]  Rüdiger Weiner,et al.  The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods , 2003 .

[7]  J. Kraaijevanger.  Contractivity of Runge-Kutta methods , 1991 .

[8]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[9]  Culbert B. Laney,et al.  Computational Gasdynamics: Waves , 1998 .

[10]  Steven J. Ruuth,et al.  A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods , 2002, SIAM J. Numer. Anal..

[11]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[12]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[13]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[14]  Steven J. Ruuth,et al.  Two Barriers on Strong-Stability-Preserving Time Discretization Methods , 2002, J. Sci. Comput..

[15]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[16]  Willem Hundsdorfer,et al.  Monotonicity-Preserving Linear Multistep Methods , 2003, SIAM J. Numer. Anal..

[17]  Kevin Burrage,et al.  Efficiently Implementable Algebraically Stable Runge–Kutta Methods , 1982 .

[18]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[19]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[20]  M. N. Spijker.  Contractivity in the numerical solution of initial value problems , 1983 .

[21]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[22]  Chi-Wang Shu,et al.  A Survey of Strong Stability Preserving High Order Time Discretizations , 2001 .

[23]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .