Energy Stability of Explicit Runge-Kutta Methods for Non-autonomous or Nonlinear Problems

TL;DR

This paper analyzes the energy stability of explicit Runge-Kutta methods for nonlinear and non-autonomous problems, providing conditions for their conditional and unconditional energy stability, and constructing examples of such schemes.

Contribution

It offers necessary and sufficient conditions for energy stability of explicit Runge-Kutta methods in complex problems, including examples of conditionally and unconditionally stable schemes.

Findings

Conditions for energy stability are established.

Explicit schemes with unconditional energy stability are constructed.

Examples demonstrate the practical application of the theoretical results.

Abstract

Many important initial value problems have the property that energy is non-increasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge-Kutta methods for nonlinear or non-autonomous problems. We provide both necessary and sufficient conditions for energy stability over these classes of problems. Examples of conditionally energy stable schemes are constructed and an example is given in which unconditional energy stability is obtained with an explicit scheme.