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Validity of relaxation models arising from numerical schemes for hyperbolic-parabolic systems

Zhiting Ma, Weifeng Zhao

TL;DR

This work addresses the validity of relaxation models that arise from numerical schemes for hyperbolic-parabolic systems by applying the general convergence framework developed in Peng 2025. It verifies convergence criteria for five representative relaxation models—covering direct convective/diffusive relaxations, lattice Boltzmann, and diffusive kinetic approaches—and presents a new relaxation model valid for general multi-dimensional HP systems. The results establish that the relaxation solutions converge to the target hyperbolic-parabolic limit as the relaxation parameter $\varepsilon$ tends to zero, under mild structural assumptions. This provides a rigorous foundation for the use of relaxation-based schemes in multiscale dissipative PDEs and broadens the toolbox with a generally applicable model.

Abstract

This work is concerned with relaxation models arising from numerical schemes for hyperbolic-parabolic systems. Such models are a hyperbolic system with both the hyperbolic part and the stiff source term involving a small positive parameter, and thus are endowed with complicated multiscale properties. Relaxation models are the basis of constructing corresponding numerical schemes and a critical issue is the convergence of their solutions to those of the given target systems, the justification of which is still lacking. In this work, we employ the recently proposed theory for general hyperbolic relaxation systems to validate relaxation models in numerical schemes of hyperbolic-parabolic systems. By verifying the convergence criteria, we demonstrate the convergence, and thereby the approximation validity, of five representative relaxation models, providing a solid basis for the effectiveness of the corresponding numerical schemes. Moreover, we propose a new relaxation model for the general multi-dimensional hyperbolic-parabolic system. With some mild assumptions on the system, we show that the proposed model satisfies the convergence criteria. We remark that the existing relaxation models are constructed only for a special case of hyperbolic-parabolic system, while our new relaxation model is valid for general systems.

Validity of relaxation models arising from numerical schemes for hyperbolic-parabolic systems

TL;DR

This work addresses the validity of relaxation models that arise from numerical schemes for hyperbolic-parabolic systems by applying the general convergence framework developed in Peng 2025. It verifies convergence criteria for five representative relaxation models—covering direct convective/diffusive relaxations, lattice Boltzmann, and diffusive kinetic approaches—and presents a new relaxation model valid for general multi-dimensional HP systems. The results establish that the relaxation solutions converge to the target hyperbolic-parabolic limit as the relaxation parameter tends to zero, under mild structural assumptions. This provides a rigorous foundation for the use of relaxation-based schemes in multiscale dissipative PDEs and broadens the toolbox with a generally applicable model.

Abstract

This work is concerned with relaxation models arising from numerical schemes for hyperbolic-parabolic systems. Such models are a hyperbolic system with both the hyperbolic part and the stiff source term involving a small positive parameter, and thus are endowed with complicated multiscale properties. Relaxation models are the basis of constructing corresponding numerical schemes and a critical issue is the convergence of their solutions to those of the given target systems, the justification of which is still lacking. In this work, we employ the recently proposed theory for general hyperbolic relaxation systems to validate relaxation models in numerical schemes of hyperbolic-parabolic systems. By verifying the convergence criteria, we demonstrate the convergence, and thereby the approximation validity, of five representative relaxation models, providing a solid basis for the effectiveness of the corresponding numerical schemes. Moreover, we propose a new relaxation model for the general multi-dimensional hyperbolic-parabolic system. With some mild assumptions on the system, we show that the proposed model satisfies the convergence criteria. We remark that the existing relaxation models are constructed only for a special case of hyperbolic-parabolic system, while our new relaxation model is valid for general systems.
Paper Structure (10 sections, 3 theorems, 87 equations)

This paper contains 10 sections, 3 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Convergence criteria
  3. Validity of relaxation models
  4. Relaxation model for 1D CDE
  5. Relaxation model for viscous conservation law
  6. Relaxation model for multidimensional nonlinear diffusion equation
  7. Lattice Boltzmann model for CDE
  8. Diffusive kinetic model for nonlinear parabolic systems
  9. A new relaxation model
  10. Conclusions and remarks

Key Result

Theorem 2.1

Denote $\tilde{U}(x, \varepsilon)=(\tilde{U}^{I}(x, \varepsilon),\tilde{U}^{II}(x, \varepsilon))^T$ and $u_0(x, 0)$ as the initial data for the relaxation system equ:2-1-pde and the limit equation 4eq:u0-equ, respectively, which are assumed to satisfy the consistency condition Assume Conditions cond:2-ssc-1-cond:2-ssc-5 hold. Then there exists a positive constant $T_\star$, independent of $\varep

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Remark 4.3