High-order linear multistep methods with general monotonicity and boundedness properties

We consider linear multistep methods that possess general monotonicity and boundedness properties. Strict monotonicity, 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 with the strong-stability-preserving (SSP) Runge–Kutta methods. By relaxing these strict monotonicity requirements a larger class of methods can be considered, including many methods of practical interest. s 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 monotonicity-preserving multistep schemes. Moreover there is a substantial gain in efficiency compared to recently constructed SSP Runge–Kutta (SSPRK) methods.