Optimal monotonicity-preserving perturbations of a given Runge-Kutta method
This work addresses the problem of improving monotonicity preservation in numerical ODE solvers, which is important for applications like hyperbolic PDEs, but the results are incremental as they extend known perturbation techniques.
The paper studies how to optimally perturb a given Runge-Kutta method to increase its radius of absolute monotonicity, proving that methods with zero radius can always be perturbed to have positive radius, and providing algorithms for computing optimal perturbations.
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.