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Optimal monotonicity-preserving perturbations of a given Runge-Kutta method. (English) Zbl 1397.65103
Summary: Perturbed Runge-Kutta methods (also referred to as downwind Runge-Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge-Kutta counterparts. In this paper we study the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.
MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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