Krylov SSP Integrating Factor Runge-Kutta WENO Methods

Abstract

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numeri-cally solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear com-ponent and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method. Keywords: strong stability preserving; integrating factor; Runge–Kutta; weighted essentially non-oscillatory methods; Krylov subspace approximation