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. 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 optimal 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 are achieved.

翻译：本文聚焦于非线性退化抛物方程的有限差分逼近，此类抛物方程的特点在于粘性项在某些区域消失。这种消失性除了带来抛物方程中显式时间离散化常见的严格CFL条件限制外，还产生了捕捉陡峭前沿的额外挑战。为精确解析陡峭前沿，我们采用高阶多分辨率交替有限差分WENO（A-WENO）方法进行空间离散。为缓解非线性刚性扩散项导致的时间步长限制，我们采用指数时间差分龙格-库塔（ETD-RK）方法——一类高效精确的指数积分器——进行时间离散。然而，对于高阶WENO格式这类高度非线性的空间离散化，如何在应用指数积分器时高效构造线性刚性部分仍具挑战，因为直接计算非线性扩散项高阶WENO离散化的雅可比矩阵极其复杂且计算代价高昂。本文基于以下事实提出一种新颖有效的解决方案：在光滑区域非线性权重近似于最优线性权重，而在非光滑区域刚性扩散项会发生退化。据此，我们在ETD-RK时间推进中用对应高阶线性格式的雅可比矩阵替代高阶多分辨率A-WENO格式的精确雅可比矩阵。文中详述了算法设计，并通过大量数值实验验证了该处理方式的有效性及本方法的优越性能。该方法能有效处理非线性抛物型偏微分方程的刚性，并实现了大时间步长计算。