MARS : a Method for the Adaptive Removal of Stiffness in PDEs

TL;DR

The paper introduces MARS, an adaptive method that dynamically adjusts an operator to suppress numerical instabilities in PDEs, achieving implicit stability with explicit-like computational efficiency.

Contribution

It extends the EI method by developing an adaptive procedure to optimize the operator for stability, applicable to nonlinear and non-local PDEs in multiple dimensions.

Findings

Successfully stabilizes nonlinear PDEs in 1D and 2D

Automatically adapts to theoretical stability conditions

Maintains computational cost similar to explicit methods

Abstract

The E(xplicit)I(implicit)N(null) method was developed recently to remove numerical instability from PDEs, adding and subtracting an operator $\mathcal{D}$ of arbitrary structure, treating the operator implicitly in one case, and explicitly in the other. Here we extend this idea by devising an adaptive procedure to find an optimal approximation for $\mathcal{D}$. We propose a measure of the numerical error which detects numerical instabilities across all wavelengths, and adjust each Fourier component of $\mathcal{D}$ to the smallest value such that numerical instability is suppressed. We show that for a number of nonlinear and non-local PDEs, in one and two dimensions, the spectrum of $\mathcal{D}$ adapts automatically and dynamically to the theoretical result for marginal stability. Our method thus has the same stability properties as a fully implicit method, while only requiring the computational cost comparable to an explicit solver. The adaptive implicit part is diagonal in Fourier space, and thus leads to minimal overhead compared to the explicit method.