We consider linear multistep methods that possess the TVD (total variation diminishing) or TVB (total variation bounded) properties, or related general monotonicity and boundedness properties. Strict monotonicity or TVD, in terms of arbitrary starting values for the multistep schemes, is only valid for a small class of methods, under very stringent step size restrictions. This makes them uncompetitive to the TVD Runge-Kutta methods. By relaxing these strict monotonicity requirements a larger class of methods can be considered, including many methods of practical interest. In this paper we construct linear multistep methods of high-order (up to six) that possess relaxed monotonicity or boundedness properties with optimal step size conditions. Numerical experiments show that the new schemes perform much better than the classical TVD multistep schemes. Moreover there is a substantial gain in efficiency compared to recently constructed TVD Runge-Kutta methods.

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Modelling, Analysis and Simulation [MAS]
Multiscale Dynamics

Ruuth, S. J., & Hundsdorfer, W. (2004). High-order TVD and TVB linear multistep methods. Modelling, Analysis and Simulation [MAS]. CWI.