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A High Order IMEX Method for Generalized Korteweg de-Vries Equations

Seth Gerberding

TL;DR

The paper develops a robust high-order IMEX method for generalized Korteweg de-Vries equations by coupling $H^1$-conforming finite elements in space with a split IMEX time integrator that treats the dispersive third-order term implicitly and the nonlinear hyperbolic flux explicitly. A reformulation with auxiliary variables reduces spatial derivatives to second order, enabling $H^1$ discretization while preserving mass and achieving $L^2$-stability; high-order accuracy is attained through graph-viscosity, a limiter, consistent mass, and a high-order ERK-DIRK time integration. Theoretical results include $\ell^2$ stability and mass conservation, plus well-posed dispersive updates, complemented by extensive numerical simulations (convergence tests, Zabusky-Kruskal, and multi-soliton interactions) validating accuracy and robustness. The method offers a CFL requirement of $\tau = \mathcal{O}(h)$, avoids nonlinear solves, and generalizes to polynomial and system fluxes, indicating strong practical significance for simulating dispersive wave phenomena with general flux structures.

Abstract

In this paper, we introduce a high order space-time approximation of generalized Korteweg de-Vries equations. More specifically, the method uses continuous $H^1$-conforming finite elements for the spatial approximation and implicit-explicit methods for the temporal approximation. The method is high order in both space, provably stable, and mass-conservative. The scheme is formulated, its properties are proven, and numerical simulations are provided to illustrate the proposed methodology.

A High Order IMEX Method for Generalized Korteweg de-Vries Equations

TL;DR

The paper develops a robust high-order IMEX method for generalized Korteweg de-Vries equations by coupling -conforming finite elements in space with a split IMEX time integrator that treats the dispersive third-order term implicitly and the nonlinear hyperbolic flux explicitly. A reformulation with auxiliary variables reduces spatial derivatives to second order, enabling discretization while preserving mass and achieving -stability; high-order accuracy is attained through graph-viscosity, a limiter, consistent mass, and a high-order ERK-DIRK time integration. Theoretical results include stability and mass conservation, plus well-posed dispersive updates, complemented by extensive numerical simulations (convergence tests, Zabusky-Kruskal, and multi-soliton interactions) validating accuracy and robustness. The method offers a CFL requirement of , avoids nonlinear solves, and generalizes to polynomial and system fluxes, indicating strong practical significance for simulating dispersive wave phenomena with general flux structures.

Abstract

In this paper, we introduce a high order space-time approximation of generalized Korteweg de-Vries equations. More specifically, the method uses continuous -conforming finite elements for the spatial approximation and implicit-explicit methods for the temporal approximation. The method is high order in both space, provably stable, and mass-conservative. The scheme is formulated, its properties are proven, and numerical simulations are provided to illustrate the proposed methodology.

Paper Structure

This paper contains 25 sections, 9 theorems, 45 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The model problem
  3. Low-Order Approximation details
  4. Temporal Discretization
  5. Reformulation
  6. Spatial Discretization
  7. Explicit Hyperbolic Prediction
  8. Implicit Dispersive Update
  9. Euler-IMEX
  10. High Order
  11. High Order Hyperbolic Prediction
  12. High Order Dispersive Update
  13. High order in time
  14. Hyperbolic Prediction
  15. Dispersive Update
  16. ...and 10 more sections

Key Result

Lemma 3.1

Let $(u,z,g)$ be a smooth periodic solution to eqn: disp-reform-u-eqn: disp-reform-g. Then the following energy conservation equation holds:

Figures (5)

  • Figure 1: Comparison of the IMEX method to the Leap-Frog method of Zabusky_65.
  • Figure 2: Comparison of the IMEX method to the TDSR method of Chandramouli_Farhat_Musslimani_22.
  • Figure 3: Space time plot for the single soliton solution \ref{['eqn: kdv-solution']} of the KdV equation \ref{['eqn: kdv']} with initial condition $u_0(x) = 2\mathop{\mathrm{sech}}\nolimits^2(x)$.
  • Figure 4: Space-time plots of the two-soliton interacting solution \ref{['eqn: kdv-2-soliton-solution']} for the KdV equation \ref{['eqn: kdv']}. Left: surface plot. Right: contour plot.
  • Figure 5: Space-time plots for the three-soliton interaction for the KdV equation \ref{['eqn: kdv']}. Left: Surface plot. Right: contour plot.

Theorems & Definitions (25)

  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Corollary 3.3: Hyperbolic stability
  • Proof 3
  • Lemma 3.4: Hyperbolic mass conservation
  • Proof 4
  • Lemma 3.5: Dispersive stability
  • Proof 5
  • ...and 15 more