Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
SIAM Journal on Numerical Analysis
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In the literature, on the numerical solution of nonlinear time dependent partial differential equations, much attention has been paid to numerical processes which have the favourable property of being total variation bounded (TVB). A popular approach to guaranteeing the TVB property consists in demanding that the process has the stronger property of being total variation diminishing (TVD).For Runge-Kutta methods--applied to semi-discrete approximations of partial differential equations--conditions on the time step were established which guarantee the TVD property; see, e.g., [J. Comput. Phys. 77 (1988) 439; Math. Comp. 67 (1998) 73; SIAM Rev. 43 (2001) 89; SIAM J. Numer. Anal. (2002), in press; Higueras, Tech. Report, Universidad Pública de Navarra, 2002; SIAM J. Numer. Anal. 40 (2002) 469]. These conditions were derived under the assumption that the simple explicit Euler time stepping process is TVD.However, for various important semi-discrete approximations, the Euler process is TVB but not TVD--see, e.g., [Math. Comp. 49 (1987) 105; Math. Comp. 52 (1989) 411]. Accordingly, the above stepsize conditions for Runge-Kutta methods are not directly relevant to such approximations, and there is a need for stepsize restrictions with a wider range of applications.In this paper, we propose a general theory yielding stepsize restrictions which cover a larger class of semi-discrete approximations than covered thus far in the literature. In particular, our theory gives stepsize restrictions, for general Runge-Kutta methods, which guarantee total-variation-boundedness in situations where the Euler process is TVB but not TVD.