High-order exponential time differencing multi-resolution alternative finite difference WENO methods for nonlinear degenerate parabolic equations

In this paper, we focus on the finite difference approximation of nonlinear degenerate parabolic equations, a special class of parabolic equations where the viscous term vanishes in certain regions. This vanishing gives rise to additional challenges in capturing sharp fronts, beyond the restrictive CFL conditions commonly encountered with explicit time discretization in parabolic equations. To resolve the sharp front, we adopt the high-order multi-resolution alternative finite difference WENO (A-WENO) methods for the spatial discretization, which is designed to effectively suppress oscillations in the presence of large gradients and achieve nonlinear stability. To alleviate the time step restriction from the nonlinear stiff diffusion terms, we employ the exponential time differencing Runge-Kutta (ETD-RK) methods, a class of efficient and accurate exponential integrators, for the time discretization. However, for highly nonlinear spatial discretizations such as high-order WENO schemes, it is a challenging problem how to efficiently form the linear stiff part in applying the exponential integrators, since direct computation of a Jacobian matrix for high-order WENO discretizations of the nonlinear diffusion terms is very complicated and expensive. Here we propose a novel and effective approach of replacing the exact Jacobian of high-order multi-resolution A-WENO scheme with that of the corresponding high-order linear scheme in the ETD-RK time marching, based on the fact that in smooth regions the nonlinear weights closely approximate the corresponding linear weights, while in non-smooth regions the stiff diffusion degenerates. The algorithm is described in detail, and numerous numerical experiments are conducted to demonstrate the effectiveness of such a treatment and the good performance of our method. The stiffness of the nonlinear parabolic partial differential equations (PDEs) is resolved well, and large time-step size computations of Δ t ∼ O ( Δ x ) are achieved. • A novel and effective semilinearization approach is proposed to solve the difficult problem of how to efficiently apply exponential integrators, e.g. the exponential time differencing methods, to high-order WENO schemes for solving stiff nonlinear degenerate parabolic PDEs. • The spatial discretizations are based on the high-order multi-resolution alternative finite difference WENO (A-WENO) methods. They are flexible in constructing linear weights, which simplifies the procedure in applications of WENO schemes to solve degenerate parabolic PDEs. • The ETD-RK methods resolve the stiffness of the nonlinear degenerate parabolic PDEs very well, and the desired large time-step size simulations of Δ t O ( Δ x ) are achieved. • Extensive numerical experiments are performed to demonstrate the effectiveness of this new semilinearization approach for exponential integrators, and verify high-order accuracy, nonlinear stability and high efficiency of the developed ETD-RK multi-resolution A-WENO methods. • The numerical comparisons with some commonly used explicit and implicit SSP-RK methods show that the proposed methods are more efficient in solving the nonlinear degenerate parabolic PDEs, especially the multidimensional problems.