2019/04/15 by Leah Isherwood, Isherwood, Leah, Zachary J. Grant +3
Mathematics · Engineering · Computer Science · #Numerical methods for differential equations #Advanced Numerical Methods in Computational Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1904.07194
Problems that feature significantly different time scales, where the stiff\ntime-step restriction comes from a linear component, implicit-explicit (IMEX)\nmethods alleviate this restriction if the concern is linear stability. However,\nwhere the SSP property is needed, IMEX SSP Runge-Kutta (SSP-IMEX) methods have\nvery restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an\nintegrating factor approach to handle the linear component exactly and step the\ntransformed problem forward using some time-evolution method. The strong\nstability properties of integrating factor Runge--Kutta methods were previously\nestablished, where it was shown that it is possible to define explicit\nintegrating factor Runge-Kutta methods that preserve strong stability\nproperties satisfied by each of the two components when coupled with forward\nEuler time-stepping. It was proved that the solution will be SSP if the\ntransformed problem is stepped forward with an explicit SSP Runge-Kutta method\nthat has non-decreasing abscissas. However, explicit SSP Runge-Kutta methods\nhave an order barrier of p=4, and sometimes higher order is desired. In this\nwork we consider explicit SSP two-step Runge--Kutta integrating factor methods\nto raise the order. We show that strong stability is ensured if the two-step\nRunge-Kutta method used to evolve the transformed problem is SSP and has\nnon-decreasing abscissas. We find such methods up to eighth order and present\ntheir SSP coefficients. Adding a step allows us to break the fourth order\nbarrier on explicit SSP Runge-Kutta methods; furthermore, our explicit SSP\ntwo-step Runge--Kutta methods with non-decreasing abscissas typically have\nlarger SSP coefficients than the corresponding one-step methods.\n