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It is difficult to design high order numerical schemes which could preserveboth the maximum bound property (MBP) and energy dissipation law for certainphase field equations. Strong stability preserving (SSP) Runge-Kutta methodshave been developed for numerical solution of hyperbolic partial differentialequations in the past few decades, where strong stability means thenon-increasing of the maximum bound of the underlying solutions. However,existing framework of SSP RK methods can not handle nonlinear stabilities likeenergy dissipation law. The aim of this work is to extend this SSP theory todeal with the nonlinear phase field equation of the Allen-Cahn type whichtypically satisfies both maximum bound preserving (MBP) and energy dissipationlaw. More precisely, for Runge-Kutta time discretizations, we first derive ageneral necessary and sufficient condition under which MBP is satisfied; and wefurther provide a necessary condition under which the MBP scheme satisfiesenergy dissipation.

Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equation