High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations

Abstract

New high-order finite difference multi-resolution weighted essentially non-oscillatory (WENO) schemes with the Lax-Wendroff (LW) time discretization method for solving fractional differential equations with the Caputo fractional derivative are considered. The fractional derivative of order $α ∈ (1, 2)$ is split into an integral part and a second derivative term. The Gauss-Jacobi quadrature method is employed to solve the integral part, and a new multi-resolution WENO method for discretizing the second derivative term is developed. High-order spatial reconstruction procedures use any positive numbers (with a minor restriction) as linear weights. The new multi-resolution WENO-LW methods are one-step explicit high-order finite difference schemes. They are more compact than multi-resolution WENO-RK schemes of the same order. The LW time discretization is more cost efficient than the Runge-Kutta time discretization method. The construction of new multi-resolution WENO-LW schemes is simple and easy to generalize to arbitrary high-order accuracy in multi-dimensions. One- and two-dimensional examples with strong discontinuities verify the good performance of new fourth-, sixth-, and eighth-order multi-resolution WENO-LW schemes.