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

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 monotonicity-preserving multistep schemes. Moreover there is a substantial gain in efficiency compared to recently constructed SSP Runge-Kutta (SSPRK) methods.