Energy-superconvergent Explicit Runge--Kutta Time Discretizations

TL;DR

This paper develops explicit Runge--Kutta methods with energy conservation properties that significantly exceed their formal order, validated on various physical and mathematical models.

Contribution

It introduces a framework for constructing energy-superconvergent RK methods with orders up to eleven, including new algorithms like RK325, RK427, and RK547, for both linear and nonlinear systems.

Findings

Energy accuracy can reach up to order 2s-p+1 for s-stage, p-th order RK methods.

New methods like RK325, RK427, and RK547 demonstrate high energy accuracy on diverse problems.

Validated performance on systems including Maxwell equations, Schrödinger, KdV, and rigid body dynamics.

Abstract

This paper investigates the energy conservation properties of explicit Runge--Kutta (RK) time discretizations for autonomous skew-symmetric systems. For linear problems, we present a general framework for constructing RK methods in which the energy-accuracy order significantly exceeds the number of stages. Specifically, for an $s$-stage, $p$-th order RK method (where $p$ is even), we prove that the energy accuracy can reach up to order $2s-p+1$. Utilizing this framework, we derive several energy-superconvergent methods, including five- to seven-stage algorithms with energy accuracy up to the eleventh order, and establish their corresponding strong stability criteria. The methods are validated on a range of benchmark problems, including harmonic oscillators, integro-differential equations in peridynamics, and the Maxwell equations. Furthermore, we extend the energy-superconvergent framework to autonomous nonlinear systems with amplitude-dependent frequencies. By deriving fifth-order energy conditions for three-stage, second-order methods, we develop the RK325 algorithm. The performance of RK325 is demonstrated for a broad range of problems, including Euler's equations for rigid body dynamics, the nonlinear Schr\"odinger equation, the Korteweg--de Vries (KdV) equation, Burgers' equation, and the Landau--Lifshitz equation. Additionally, we develop four-stage, second-order methods (RK427) and five-stage, fourth-order methods (RK547), all of which achieve seventh-order energy accuracy for the cubic nonlinear case. Finally, the performance of RK547 method is illustrated using the nonlinear Maxwell--Kerr system.