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Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation

Sebastian Bleecke, Abhijit Biswas, David I. Ketcheson, Hendrik Ranocha, Jochen Schutz

TL;DR

This work develops asymptotic-preserving and energy-conserving numerical methods for a hyperbolic BBM approximation (BBMH) of the BBM equation. By combining SBP spatial discretizations with IMEX Runge-Kutta time-stepping and a time-relaxation mechanism, the authors preserve the Hamiltonian-relative-equilibrium structure, ensure discrete energy conservation, and achieve AP behavior as $\varepsilon \to 0$. They provide a careful splitting of the BBMH system, prove AP properties for Type I and Type II IMEX schemes under GSA/ARS assumptions, and validate these results with numerical experiments showing improved long-time error growth and convergence to BBM. The methodology is applicable to a broad class of dispersive-hyperbolic models and highlights the value of structure-preserving discretizations for dispersive wave simulations.

Abstract

We study the hyperbolic approximation of the Benjamin-Bona-Mahony (BBM) equation proposed recently by Gavrilyuk and Shyue (2022). We develop asymptotic-preserving numerical methods using implicit-explicit (additive) Runge-Kutta methods that are implicit in the stiff linear part. The new discretization of the hyperbolization conserves important invariants converging to invariants of the BBM equation. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.

Asymptotic-preserving and energy-conserving methods for a hyperbolic approximation of the BBM equation

TL;DR

This work develops asymptotic-preserving and energy-conserving numerical methods for a hyperbolic BBM approximation (BBMH) of the BBM equation. By combining SBP spatial discretizations with IMEX Runge-Kutta time-stepping and a time-relaxation mechanism, the authors preserve the Hamiltonian-relative-equilibrium structure, ensure discrete energy conservation, and achieve AP behavior as . They provide a careful splitting of the BBMH system, prove AP properties for Type I and Type II IMEX schemes under GSA/ARS assumptions, and validate these results with numerical experiments showing improved long-time error growth and convergence to BBM. The methodology is applicable to a broad class of dispersive-hyperbolic models and highlights the value of structure-preserving discretizations for dispersive wave simulations.

Abstract

We study the hyperbolic approximation of the Benjamin-Bona-Mahony (BBM) equation proposed recently by Gavrilyuk and Shyue (2022). We develop asymptotic-preserving numerical methods using implicit-explicit (additive) Runge-Kutta methods that are implicit in the stiff linear part. The new discretization of the hyperbolization conserves important invariants converging to invariants of the BBM equation. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.

Paper Structure

This paper contains 16 sections, 7 theorems, 88 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Relative equilibrium structure
  3. Traveling wave solutions
  4. Computation of BBMH solitary waves via Petviashvili's method
  5. Splitting the BBMH system for IMEX methods: asymptotic-preserving time discretizations
  6. General IMEX methods of type I and type II
  7. Splitting of the BBMH system
  8. Asymptotic preserving IMEX schemes
  9. Splitting the BBMH system for IMEX methods: discrete in space
  10. Summation-by-parts operators
  11. The BBM equation
  12. The BBMH system
  13. Numerical experiments
  14. Asymptotic preserving properties
  15. Error growth in time
  16. ...and 1 more sections

Key Result

theorem 2.1

The BBMH system eq:BBMH is a Hamiltonian PDE $\partial_t q = \mathcal{J}_\mathrm{BBMH} \delta \mathcal{H}_\mathrm{BBMH}(q)$ with relative equilibrium structure $\partial_x q = \mathcal{J}_\mathrm{BBMH} \delta \mathcal{I}_\mathrm{BBMH}(q)$, where where we recall that $\chi_x = w$.

Figures (4)

  • Figure 1: A peakon solution of the BBMH system, obtained by integrating \ref{['bbmh-traveling-odes']} with $\varepsilon^2=4/3$, $c=1/2$.
  • Figure 2: Numerical solution of the $u$ component of the BBMH system compared to the $u$ component of the analytical solution of the BBM equation.
  • Figure 3: Error growth of the system variable $q$ with respect to the Petviashvili generated solution of the BBMH system for $\varepsilon = 10^{-3}$.
  • Figure 4: Error growth of the $u$ component with respect to the analytical solution of the BBM equation $\eta$.

Theorems & Definitions (21)

  • theorem 2.1
  • remark 2.2
  • definition 4.1
  • theorem 4.2
  • proof
  • definition 4.3
  • definition 4.4
  • remark 4.5
  • theorem 4.6
  • proof
  • ...and 11 more