Linearly Stabilized Schemes for the Time Integration of Stiff Nonlinear PDEs

TL;DR

This paper introduces linearly stabilized schemes that efficiently and unconditionally stably integrate stiff nonlinear PDEs by combining explicit nonlinear term computation with linear system solutions.

Contribution

The paper presents new and existing linearly stabilized schemes that are unconditionally stable for stiff nonlinear PDEs, simplifying implementation and reducing computational cost.

Findings

Schemes are unconditionally stable for stiff nonlinear PDEs.

Explicitly compute nonlinear terms while solving a fixed linear system.

Applications demonstrate effectiveness of the proposed methods.

Abstract

In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this article, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute the nonlinear term explicitly and, at the cost of solving a linear system with a matrix that is fixed throughout, are unconditionally stable, thus combining the advantages of explicit and implicit methods. Applications are presented to illustrate the use of these methods.