Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations

Abstract

In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory (WENO) methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicates that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fast O(N ) summation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings.

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

“…where µ = e −α(b−a) . Hence, following the idea in [10], when φ is a periodic function, we can approximate the first derivative φ ± x with (modified) partial sums in (2.11),…”

“…Note that there is an extra term for k = 3. As remarked in [10], such a term is needed for attaining unconditional stability of the scheme. An error estimate for the approximation (2.14) regarding the truncation of the infinite sum, carried out in [10], showed that keeping k terms of the partial sums led to…”

“…We start with a brief review on construction of derivative operators using the methodologies proposed in [10,11]. The second order derivative, e.g., ∂ xx , shall described first, as it will be used in the representation of first derivatives.…”

“…As part of this embedding process, a kernel parameter β was introduced, and through a careful selection, was shown to yield schemes which are A-stable. Remarkably, it was shown that one could couple these representations for the derivative operators with an explicit time-stepping method, such as the strong-stability-preserving Runge-Kutta (SSP-RK) methods [17] and still obtain schemes which maintain unconditional stability [10,11]. To address shock-capturing and control non-physical oscillations, the latter two papers introduced quadrature formulas based on WENO reconstruction, along with a nonlinear filter.This paper seeks to extend the work in [10,11] to the H-J equations (1.1) defined on non-uniformly distributed spatial domains.…”

“…This paper seeks to extend the work in [10,11] to the H-J equations (1.1) defined on non-uniformly distributed spatial domains. In particular, several improvements are given.…”