Explicit Monotone Stable Super-Time-Stepping Methods for Finite Time Singularities

TL;DR

This paper introduces explicit super-time-stepping methods, RKL and RKG, for efficiently resolving finite-time singularities in nonlinear PDEs, offering stability, accuracy, and computational advantages over traditional implicit methods.

Contribution

The paper develops and analyzes explicit RKL and RKG super-time-stepping methods that are monotone and stable, specifically tailored for resolving finite-time singularities in nonlinear PDEs.

Findings

RKL and RKG methods achieve smaller run times compared to implicit methods.

Numerical monotonicity is proven for RKL and RKG under certain conditions.

Methods maintain accuracy while resolving singularity scales without smaller timesteps.

Abstract

We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge--Kutta--Legendre (RKL) and Runge--Kutta--Gegenbauer (RKG) super-time-stepping methods were originally developed for nonlinear complex physics problems with diffusion. These are multi-stage single step second-order, forward-in-time methods with no implicit solves. The advantage is that the timestep size for stability scales with stage number $s$ as $\mathcal{O}(s^2)$. Many interesting nonlinear PDEs have finite-time singularities, and the presence of diffusion often limits one to using implicit or semi-implicit timestep methods for stability constraints. Finite-time singularities are particularly challenging due to the large range of scales that one desires to resolve, often with adaptive spatial grids and adaptive timesteps. Here we show two examples of nonlinear PDEs for which the self-similar singularity structure has time and space scales that are resolvable using the RKL and RKG methods, without forcing even smaller timesteps. Compared to commonly-used implicit numerical methods, we achieve significantly smaller run time while maintaining comparable accuracy. We also prove numerical monotonicity for both the RKL and RKG methods under their linear stability conditions for the constant coefficient heat equation, in the case of infinite domain and periodic boundary condition, leading to a theoretical guarantee of the superiority of the RKL and RKG methods over traditional super-time-stepping methods, such as the Runge-Kutta-Chebyshev (RKC) and the orthogonal Runge-Kutta-Chebyshev (ROCK) methods. Code can be found at https://github.com/ZT220501/SRK-Singularity.