Energy Conservative Relaxation-Free Runge-Kutta Schemes

TL;DR

The paper tackles the challenge of preserving energy-like invariants in time integration of semi-discrete PDEs. It introduces Relaxation-Free Runge-Kutta (RF-RK), a modification of explicit RK schemes that enforces energy conservation without relaxing the time step or sacrificing the base method’s order. Theoretical results guarantee the existence of a real correction parameter $\epsilon_n$ for small $\Delta t$, with $\epsilon_n = O(\Delta t^{p-1})$, and show the RF-RK update preserves the original order $p$ while keeping the energy change at each step zero up to machine precision. Numerical experiments across linear, linear energy-decaying, nonlinear, and Burgers’ equation–type problems demonstrate robust energy conservation, fixed-step performance, and comparable or improved accuracy relative to standard RK methods. Overall, RF-RK offers a practical, low-cost framework for energy-conservative time integration in PDE solvers with broad applicability to nonlinear stability analyses.

Abstract

A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under the exact solution of their governing PDEs. However, standard temporal schemes, such as classical Runge-Kutta (RK) methods, do not enforce these constraints, leading to a loss of accuracy and stability. Previously, the Incremental Directional Technique RK (IDT-RK) and Relaxation Runge-Kutta (R-RK) approaches have been proposed to address this. However, these lead to a loss of accuracy in the case of IDT-RK, or a loss of step size control in the case of R-RK. In the current work we propose Relaxation-Free Runge- Kutta (RF-RK) schemes, which conserve energy, maintain order of accuracy, and maintain a constant step size, alleviating many of the limitations of the aforementioned techniques. Importantly, they do so with minimal additional computational cost compared to the base RK scheme. Numerical results demonstrate that these properties are observed in practice for a range of applications. Therefore, the proposed RF-RK framework is a promising approach for energy conservative time integration of systems of PDEs.

Figures (10)

• Figure 1: Change in linear stability region for integration schemes after application of RF-RK method for $\epsilon_n$ between $-0.05$ to $0.05$, where the black lines belong to $\epsilon_n=0$, and the darker blue lines belong to positive values for $\epsilon_n$. Note that these values for $\epsilon_n$ are chosen large to exhibit its dependence clearly.
• Figure 2: Relative amplification for each wavelength when using white noise initial data for the problem \ref{['ex1_ode']}. It is integrated up to $t_f=1$, with the step sizes $\Delta t= \mu \Delta t_{max}$. While with the RK(4,4) the total energy has decreased, the two energy conserving schemes amplifyed lower modes to cancel out the energy loss by high wavenumber modes.
• Figure 3: Relative amplification for each wavelength with smooth initial data for the problem \ref{['ex1_ode']}. Integration is performed up to $t_f=1$, with the step sizes $\Delta t= \mu \Delta t_{max}$. Since most of the initial energy is distributed among the low wavenumber modes, energy change with RK(4,4) is small, and the curves for RK(4,4) and the energy conserving schemes are similar.
• Figure 4: Final solutions for problem \ref{['ex1_ode']} with the smooth initial data, integrated up to $t_f=400\pi$, with $\mu=0.99$. The two energy conserving schemes, R-RK(4,4) and RF-RK(4,4) behave similarly for this step size.
• Figure 5: Comparison of the solutions out of the two energy conserving schemes for the problem \ref{['ex1_ode']} when it is integrated up to $t_f=400\pi$ with $\mu= 0.99$, and $\mu= 1.0001$. While $\Delta t= 1.0001 \Delta t_{max}$ is slightly higher than the stability limit for R-RK(4,4), it is still within the stability limit for RF-RK(4,4).

Theorems & Definitions (6)

• Lemma 1
• proof : Proof of Lemma \ref{['Delta_positive']}
• Lemma 2
• proof : Proof of Lemma \ref{['epsilon_order']}
• Theorem 3
• proof : Proof of Theorem \ref{['RFRK_order']}