Multiple-Relaxation Runge Kutta Methods for Conservative Dynamical Systems

TL;DR

This paper introduces multiple-relaxation Runge-Kutta methods that preserve several nonlinear invariants in dynamical systems, enhancing accuracy and stability in simulations like the Korteweg-de Vries equation.

Contribution

It generalizes relaxation time stepping to conserve multiple invariants using embedded Runge-Kutta methods, with proofs of accuracy and parameter existence.

Findings

Successfully applied to Korteweg-de Vries equation

Conserved multiple invariants in multi-soliton solutions

Demonstrated computational efficiency and accuracy

Abstract

We generalize the idea of relaxation time stepping methods in order to preserve multiple nonlinear conserved quantities of a dynamical system by projecting along directions defined by multiple time stepping algorithms. Similar to the directional projection method of Calvo et. al., we use embedded Runge-Kutta methods to facilitate this in a computationally efficient manner. Proof of the accuracy of the modified RK methods and the existence of valid relaxation parameters are given, under some restrictions. Among other examples, we apply this technique to Implicit-Explicit Runge-Kutta time integration for the Korteweg-de Vries equation and investigate the feasibility and effect of conserving multiple invariants for multi-soliton solutions.