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Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions

Jan Giesselmann, Hendrik Ranocha

Abstract

We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings.

Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions

Abstract

We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings.

Paper Structure

This paper contains 19 sections, 5 theorems, 110 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Dispersive equations with mixed space-time derivatives
  3. Basic idea of the convergence analysis
  4. Convergence analysis
  5. Numerical demonstration
  6. Dispersive equations with purely spatial higher-order derivatives
  7. Odd leading order m
  8. Korteweg-de Vries equation
  9. Korteweg-de Vries-Burgers equation
  10. Gardner equation
  11. Kawahara equation
  12. Numerical demonstration
  13. Generalized Kawahara equation
  14. Relative equilibrium structure
  15. Even leading order m
  16. ...and 4 more sections

Key Result

theorem 2.1

Let $T>0$ and $f \in W^{2,\infty}_\mathrm{loc}(\mathbb{R})$ such that $f" \in L^\infty(\mathbb{R})$. Let $u \in H^{4}((0,T) \times \Omega)$ such that $\partial_x u \in L^\infty((0,T)\times \Omega)$ be a solution to eq:mixed_limit with initial data $u_0 \in H^{4}(\Omega)$. Let, for each $\tau>0$, $q$

Figures (9)

  • Figure 1: Convergence of the hyperbolic approximation \ref{['eq:bbmh_semidiscretization']} to the BBM equation \ref{['eq:bbm']}. The left plot shows the numerical solution at the final time $T = 100$ and the initial condition. The right plot shows the convergence of the discrete $L^2$ error at the final time $T$ as a function of $\tau$.
  • Figure 2: Convergence of the hyperbolic approximation \ref{['eq:spatial_odd_hyperbolic']} with $\sigma_0 = 1$ and $\mu = 0$ to the KdV equation \ref{['eq:kdv']}. The left plot shows the numerical solution at the final time $T = 100$ and the initial condition. The right plot shows the convergence of the discrete $L^2$ error at the final time $T$ as a function of the hyperbolic relaxation parameter $\tau$.
  • Figure 3: Convergence of the hyperbolic approximation \ref{['eq:spatial_odd_hyperbolic']} with $\sigma_0 = 1$ and $\mu = 0.1$ to the KdV-Burgers equation \ref{['eq:kdv_burgers']}. The left plot shows the numerical solution at the final time $T = 100$ and the initial condition. The right plot shows the convergence of the discrete $L^2$ error at the final time $T$ as a function of the hyperbolic relaxation parameter $\tau$.
  • Figure 4: Convergence of the hyperbolic approximation to the Gardner equation \ref{['eq:gardner']} with $\sigma = 1$. The left plot shows the numerical solution at the final time $T = 83.\overline{3}$ and the initial condition. The right plot shows the convergence of the discrete $L^2$ error at the final time $T$ as a function of the hyperbolic relaxation parameter $\tau$.
  • Figure 5: Convergence of the hyperbolic approximation \ref{['eq:kawahara_hyperbolic']} to the Kawahara equation \ref{['eq:kawahara']}. The left plot shows the numerical solution at the final time $T = 657.\overline{2}$ and the initial condition. The right plot shows the convergence of the discrete $L^2$ error at the final time $T$ as a function of the hyperbolic relaxation parameter $\tau$.
  • ...and 4 more figures

Theorems & Definitions (10)

  • theorem 2.1
  • remark 2.2
  • remark 3.1
  • theorem 3.2
  • theorem 3.3
  • remark 3.4
  • theorem 3.5
  • remark 3.6
  • theorem 3.7
  • remark 3.8