A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integrators

Abstract

This paper develops a new explicit semi-analytical approach to solving PDEs based on the variation-of-constants formula, or the equivalent integration equation. These new schemes avoid the discretization of spatial derivatives. Therefore, the accuracy of the semi-analytical explicit scheme depends entirely on time integration. We first establish a semi-analytical formula for wave partial differential equations and show the consistency of the semi-analytical formula with different boundary conditions. We then present semi-analytical explicit RKN-type integrators for solving wave PDEs. This methodology is also extended to dealing with first-order hyperbolic PDEs and parabolic PDEs, and for both of them the corresponding semi-analytical explicit RK-type integrators are derived as well. Numerical simulations are implemented and the results show the effectiveness of the new semi-analytical explicit schemes. Meanwhile, an operator-variation-of-constants formula with applications is introduced for high-dimensional nonlinear wave equations before presenting the idea of semi-analytical explicit RKN integrators.