Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On hyperbolic approximations for a class of dispersive and diffusive-dispersive equations

Rahul Barthwal, Firas Dhaouadi, Christian Rohde

TL;DR

This work develops first-order hyperbolic and second-order hyperbolic-parabolic relaxations to approximate dispersive and diffusive-dispersive scalar equations with nonlinear fluxes. It establishes hyperbolicity, Riemann invariants, an entropy framework, and a Hamiltonian structure for the relaxations, and proves convergence to the target models using relative entropy. A semi-implicit Lax–Wendroff scheme demonstrates robust numerical performance for solitons, dispersive shocks, and undercompressive waves across KdV-type systems. Theoretical results are complemented by comprehensive numerical experiments on KdV, Gardner, and related models, highlighting the practical potential of energy-consistent hyperbolic relaxations for complex wave phenomena.

Abstract

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth solutions is achieved by constructing a unique symmetrizer that applies to arbitrary smooth fluxes. Under stronger conditions on the fluxes, we provide a strictly convex entropy for the hyperbolic system that corresponds to the energy of the underlying dispersive equation. To approximate diffusive-dispersive equations, we rely on a viscoelastic damped system that is compatible with the found entropy for the hyperbolic approximation of the dispersive evolution. For the resulting hyperbolic-parabolic approximation, we provide a global well-posedness result. Using the relative entropy framework \cite{dafermos2005hyperbolic}, we prove that the solutions of the approximate systems converge to solutions of the original equations. The structure of the new approximate systems allows to apply standard numerical simulation methods from the field of hyperbolic balance laws. We confirm the convergence of our approximations even beyond the validity range of our theoretical findings on set of test cases covering different target equations. We show the applicability of the approach for strong nonlinear effects leading to oscillating or shock-layer-forming behavior.

On hyperbolic approximations for a class of dispersive and diffusive-dispersive equations

TL;DR

This work develops first-order hyperbolic and second-order hyperbolic-parabolic relaxations to approximate dispersive and diffusive-dispersive scalar equations with nonlinear fluxes. It establishes hyperbolicity, Riemann invariants, an entropy framework, and a Hamiltonian structure for the relaxations, and proves convergence to the target models using relative entropy. A semi-implicit Lax–Wendroff scheme demonstrates robust numerical performance for solitons, dispersive shocks, and undercompressive waves across KdV-type systems. Theoretical results are complemented by comprehensive numerical experiments on KdV, Gardner, and related models, highlighting the practical potential of energy-consistent hyperbolic relaxations for complex wave phenomena.

Abstract

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth solutions is achieved by constructing a unique symmetrizer that applies to arbitrary smooth fluxes. Under stronger conditions on the fluxes, we provide a strictly convex entropy for the hyperbolic system that corresponds to the energy of the underlying dispersive equation. To approximate diffusive-dispersive equations, we rely on a viscoelastic damped system that is compatible with the found entropy for the hyperbolic approximation of the dispersive evolution. For the resulting hyperbolic-parabolic approximation, we provide a global well-posedness result. Using the relative entropy framework \cite{dafermos2005hyperbolic}, we prove that the solutions of the approximate systems converge to solutions of the original equations. The structure of the new approximate systems allows to apply standard numerical simulation methods from the field of hyperbolic balance laws. We confirm the convergence of our approximations even beyond the validity range of our theoretical findings on set of test cases covering different target equations. We show the applicability of the approach for strong nonlinear effects leading to oscillating or shock-layer-forming behavior.

Paper Structure

This paper contains 26 sections, 9 theorems, 141 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Hyperbolic and hyperbolic-parabolic approximations
  3. A first-order approximation of the purely dispersive equation
  4. Qualitative properties of the first-order approximation
  5. Hyperbolicity, symmetrizability and local well-posedness
  6. Riemann invariants
  7. Entropy/entropy-flux pair and weak entropy solutions of \ref{['hyperbolic_system']}
  8. Least action principle and energy conservation: Hamiltonian structure of the system \ref{['hyperbolic_system']}
  9. Second-order approximations of the diffusive-dispersive equation \ref{['main']}
  10. Qualitative properties of the second-order approximation
  11. Hyperbolic-parabolic structure, energy dissipation, and local well-posedness
  12. Global well-posedness of classical solutions of \ref{['hyperbolic_parabolic_system']} for prepared initial data
  13. Hyperbolic approximations and asymptotics
  14. Dispersive asymptotics for the first-order hyperbolic system \ref{['hyperbolic_system']}
  15. Diffusive-dispersive asymptotics for the hyperbolic-parabolic system \ref{['hyperbolic_parabolic_system']}
  16. ...and 11 more sections

Key Result

Lemma 2.1

For $\alpha > \max \left| f'(u)\right|$, the hyperbolic system hyperbolic_system can be converted into an equivalent symmetric hyperbolic form. In particular, the hyperbolic system hyperbolic_system is a Friedrichs-symmetrizable system. For there exists a time $T^*\in (0, \infty)$ such that the initial value problem for hyperbolic_system and $\mathbf{U}(\cdot,0)= \mathbf{U}_0$ in $I$ is solved in

Figures (12)

  • Figure 1: Numerical results for the AIVP corresponding to \ref{['eq:IVP_1sol']}. Left: The numerical solution $u(x)$ at different times up to the final time $T=100$. Right: The time evolution of the amplitude error $e_a$ as well as the absolute $L^2$ error between the numerical solution and the exact solution to the OIVP \ref{['eq:IVP_1sol']}. $I_c=[-2,2]$ with $N=1000$ cells, $\gamma=-10^{-2}, \alpha =10^3$ and $\beta = 10^{-6}$.
  • Figure 2: Numerical results for the AIVP corresponding to \ref{['eq:IVP_2sol']}, showing the main dynamics of the two-soliton solution. The parameters are $I_c=[-15,15]$ with $N=1000$ cells. $\gamma=-1$, $\alpha =4\times10^3$, $\beta = 10^{-6}$. $T=60$ and boundary conditions are periodic.
  • Figure 3: Time evolution of the total energy error with respect to its initial value $E^{\alpha}-E^{\alpha}_0$ for different mesh sizes. The parameters are $I_c=[-15,15]$, $\gamma=-1$, $\alpha =2\times10^3$, $\beta = 10^{-8}$, $T=100$, and boundary conditions are periodic.
  • Figure 4: Numerical results for the AIVP corresponding to \ref{['eq:IVP_dsw_sech']}. The parameters are $\gamma=-10^{-4}$, $\alpha=10^3$, $\beta=10^{-7}$. The computational domain is $I_c=[-5,5]$ with $N=8000$ cells, and the final time is $T=0.4$. Periodic boundary conditions are used.
  • Figure 5: Numerical results for the AIVP corresponding to \ref{['eq:IVP_dsw_sech']}. Left: Numerical solution at $t=2$ compared with the asymptotic envelopes $\mathrm{A}_\pm$. Right: $x-t$ diagram of the numerical solution, superimposed with the asymptotic DSW bounds. The parameters are $\gamma=-10^{-4}$, $\alpha=10^3$, $\beta=10^{-7}$. The computational domain is $I_c=[-8,2]$ with $N=10000$ cells, and the simulation is run up to $T=3$. Pseudo Neumann boundary conditions are used.
  • ...and 7 more figures

Theorems & Definitions (23)

  • Lemma 2.1: Symmetrizability and local well-posedness of the initial value problem for \ref{['hyperbolic_system']}
  • proof
  • Remark 2.1
  • Lemma 2.2: Riemann invariants for the system \ref{['hyperbolic_system']}
  • proof
  • Lemma 2.3: Entropy/entropy-flux pair for \ref{['hyperbolic_system']}
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 13 more