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Energy-superconvergent Runge-Kutta Time Discretizations

Jinjie Liu, Moysey Brio

TL;DR

The paper addresses energy accuracy in explicit Runge-Kutta time discretizations for antisymmetric autonomous systems. It develops energy-superconvergent RK (ESC-RK) methods in which the energy order $r$ can exceed the method order $p$, achieving up to $r = 2s-p+1$ for even $p$ and providing concrete $s$-stage, fourth-order families such as RK(4,4,5) through RK(7,4,11). It also derives strong stability criteria and demonstrates, via harmonic oscillators, linear peridynamics, and 1D Maxwell's equations, that energy errors can be driven to machine precision while maintaining competitive computational efficiency. These ESC-RK methods offer enhanced energy conservation and robustness, making them suitable as time integrators for Yee-like spatial discretizations in energy-sensitive simulations.

Abstract

In this paper, we investigate the energy accuracy of explicit Runge-Kutta (RK) time discretization for antisymmetric autonomous linear systems and present a framework for constructing RK methods with an order of energy accuracy much greater than the number of stages. For an $s$-stage, $p$th-order RK method, we show that the energy accuracy can achieve superconvergence with an order up to $2s-p+1$ if $p$ is even. Several energy-superconvergent methods, including five- to seven-stage fourth-order methods with energy accuracy up to the eleventh order, together with their strong stability criteria, are derived. The proposed methods are examined using several applications, including second-order ordinary differential equations for harmonic oscillators, linear integro-differential equations for peridynamics, and one-dimensional Maxwell's equations of electrodynamics.

Energy-superconvergent Runge-Kutta Time Discretizations

TL;DR

The paper addresses energy accuracy in explicit Runge-Kutta time discretizations for antisymmetric autonomous systems. It develops energy-superconvergent RK (ESC-RK) methods in which the energy order can exceed the method order , achieving up to for even and providing concrete -stage, fourth-order families such as RK(4,4,5) through RK(7,4,11). It also derives strong stability criteria and demonstrates, via harmonic oscillators, linear peridynamics, and 1D Maxwell's equations, that energy errors can be driven to machine precision while maintaining competitive computational efficiency. These ESC-RK methods offer enhanced energy conservation and robustness, making them suitable as time integrators for Yee-like spatial discretizations in energy-sensitive simulations.

Abstract

In this paper, we investigate the energy accuracy of explicit Runge-Kutta (RK) time discretization for antisymmetric autonomous linear systems and present a framework for constructing RK methods with an order of energy accuracy much greater than the number of stages. For an -stage, th-order RK method, we show that the energy accuracy can achieve superconvergence with an order up to if is even. Several energy-superconvergent methods, including five- to seven-stage fourth-order methods with energy accuracy up to the eleventh order, together with their strong stability criteria, are derived. The proposed methods are examined using several applications, including second-order ordinary differential equations for harmonic oscillators, linear integro-differential equations for peridynamics, and one-dimensional Maxwell's equations of electrodynamics.
Paper Structure (7 sections, 6 theorems, 43 equations, 5 figures, 9 tables)

This paper contains 7 sections, 6 theorems, 43 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Energy-superconvergent time discretization
  3. Examples
  4. Second-order differential equation for harmonic oscillator
  5. Linear peridynamic integro-differential equation
  6. One-dimensional Maxwell's equations
  7. Conclusion

Key Result

Proposition 1

For an $s$-stage Runge-Kutta method of order $p=s$ applied to system (eq:de), the order of energy accuracy is $r = 2\lfloor s/2 \rfloor + 1$, i.e., $r=s$ if $s$ is odd and $r=s+1$ if $s$ is even.

Figures (5)

  • Figure 1: Stability regions of several second-order methods and the RK(4,4,5) method.
  • Figure 2: Stability regions of several fourth-order methods.
  • Figure 3: Time history plots (in log-log scale) of the magnitudes of relative energy deviation for harmonic oscillator simulations. The equations represent the linear fitting of the corresponding curve in the same color. Each simulation runs for 5000 steps until $T=10^3$.
  • Figure 4: Solutions of the linear peridynamic equation at (a) $t=2.5$ and (b) $t=5$ using RK(7,4,11).
  • Figure 5: Time history plots (in log-log scale) of the magnitude of the relative energy deviation for the solution to 1D Maxwell's equations using several fourth-order RK methods. $\Delta x=\frac{\lambda_0}{20}$. The Courant number is fixed at 0.5. Each simulation runs for 100,000 iterations.

Theorems & Definitions (7)

  • Proposition 1
  • Lemma 1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 1