Kernel Based High Order "Explicit" Unconditionally-Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations
This work provides a practical solution for efficiently solving challenging nonlinear degenerate advection-diffusion equations, which are common in fluid dynamics and other fields.
The authors propose a novel numerical scheme for nonlinear degenerate parabolic equations that achieves unconditional stability with explicit time stepping, enabling much larger time steps than conventional explicit methods and resulting in significant computational savings.
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 $\mathcal{O}(N)$ 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.