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Multiderivative time integration methods preserving nonlinear functionals via relaxation

Hendrik Ranocha, Jochen Schütz

TL;DR

This work targets preserving nonlinear functionals, such as invariants and entropies, in time integration of ODEs and PDEs by marrying the relaxation approach with high-order multiderivative Runge–Kutta schemes. It yields entropy-conserving or entropy-dissipating integrators by post-processing baseline steps with a scalar relaxation parameter, supported by entropy-estimation strategies based on Gauss–Lobatto quadrature and Hermite–Birkhoff or continuous-output techniques. The authors develop stability results for the relaxed multiderivative methods and demonstrate, through extensive numerical experiments on oscillators and nonlinear PDEs (e.g., Taylor–Green vortex, BBM, and KdV), that relaxation improves long-time behavior and preserves key functionals. These findings advance structure-preserving time integration, offering robust, high-order tools for nonlinear dynamics in fluids and nonlinear waves.

Abstract

We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

Multiderivative time integration methods preserving nonlinear functionals via relaxation

TL;DR

This work targets preserving nonlinear functionals, such as invariants and entropies, in time integration of ODEs and PDEs by marrying the relaxation approach with high-order multiderivative Runge–Kutta schemes. It yields entropy-conserving or entropy-dissipating integrators by post-processing baseline steps with a scalar relaxation parameter, supported by entropy-estimation strategies based on Gauss–Lobatto quadrature and Hermite–Birkhoff or continuous-output techniques. The authors develop stability results for the relaxed multiderivative methods and demonstrate, through extensive numerical experiments on oscillators and nonlinear PDEs (e.g., Taylor–Green vortex, BBM, and KdV), that relaxation improves long-time behavior and preserves key functionals. These findings advance structure-preserving time integration, offering robust, high-order tools for nonlinear dynamics in fluids and nonlinear waves.

Abstract

We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.
Paper Structure (24 sections, 7 theorems, 54 equations, 14 figures, 2 tables)

This paper contains 24 sections, 7 theorems, 54 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic ideas of multiderivative and relaxation methods
  3. Multi-derivative Runge-Kutta methods
  4. Relaxation methods
  5. Efficient application of relaxation to multiderivative methods
  6. Entropy estimates (not only) for dissipative problems
  7. Hermite-Birkhoff interpolation
  8. Continuous Runge-Kutta output
  9. Stability properties
  10. Numerical experiments
  11. Convergence experiment I: a nonlinear oscillator
  12. Error growth in time for nonlinear oscillators
  13. Convergence experiment II: Kepler's problem
  14. Error growth in time for Kepler's problem
  15. Convergence experiment III: dissipated exponential entropy
  16. ...and 9 more sections

Key Result

theorem 2.2

Consider the relaxation procedure described above for a one-step method of order $p \ge 2$ with exact value at time $t^{n}$, i.e., $u^{n+1} = u(t^{n+1}) + \mathcal{O}(\Delta t^{p+1})$. If the time step $\Delta t$ is sufficiently small, $\eta^{\mathrm{new}} = \eta\bigl( u(t^{n+1}) \bigr) + \mathcal{O there is a unique solution $\gamma = 1 + \mathcal{O}(\Delta t^{p-1})$ and the relaxation method sat

Figures (14)

  • Figure 1: Angles of $A(\alpha)$-stability for the relaxed update with fixed relaxation parameter $\gamma > 1$ for some implicit multiderivative methods.
  • Figure 2: Angles of $A(\alpha)$-stability for the relaxed update with fixed relaxation parameter $\gamma > 1$ for some implicit multiderivative methods.
  • Figure 3: Convergence results for the nonlinear oscillators using the explicit third, fourth and fifth order methods of chan2010explicit, see also Sec. \ref{['sec:ctbutcher']}. Dotted lines correspond to the relaxed schemes, straight lines to the baseline schemes.
  • Figure 4: Convergence results for the nonlinear oscillators using several implicit schemes, see also Sec. \ref{['sec:sspbutcher']} and Sec. \ref{['sec:hbbutcher']}. Straight lines correspond to the baseline schemes, while dotted and dashed lines use relaxation. For the entropy-conservative case, only dotted lines are plotted, as $\eta^{\text{new}} \equiv \eta(u^n)$ is trivial to obtain. For the entropy-dissipative case, estimating $\eta^{\text{new}}$ is an important part of the algorithm, see Sec. \ref{['sec:entropy-estimates']}. Dotted means that the estimate of $\eta^{\text{new}}$ has been obtained using a Hermite-Birkhoff interpolation of $u$, while dashed means that the estimate has been obtained using the continuous Runge-Kutta output. Please note that the latter is not possible for the SSP scheme.
  • Figure 5: Temporal evolution of the numerical error and the functional $\eta$ for the nonlinear oscillators using the explicit third, fourth and fifth order methods of chan2010explicit, see also Sec. \ref{['sec:ctbutcher']}. Top: conservative case \ref{['eq:nonlinear_osc']}, bottom: dissipative case \ref{['eq:damped_nonlinear_osc']}. Dashed lines denote schemes that use relaxation.
  • ...and 9 more figures

Theorems & Definitions (14)

  • remark 2.1
  • theorem 2.2
  • lemma 3.1
  • remark 3.2
  • theorem 4.1
  • proof
  • corollary 4.2
  • corollary 4.3
  • theorem 4.4
  • proof
  • ...and 4 more