Reconstruction of finite volume solution for parameter-dependent linear hyperbolic conservation laws

TL;DR

This paper introduces a Reconstruct-Transform-Average (RTA) numerical method for efficiently approximating solutions of parameter-dependent linear hyperbolic conservation laws, especially for discontinuous solutions, using pre-computed snapshots.

Contribution

It proposes a novel RTA algorithm that reconstructs solutions from snapshots through transformation and averaging, tailored for parameter-dependent hyperbolic PDEs, with detailed analysis and numerical validation.

Findings

The RTA method accurately approximates solutions for transport equations.

Numerical results demonstrate the method's effectiveness for elastodynamics.

The approach efficiently handles parameter variations in hyperbolic problems.

Abstract

This paper is concerned with the development of suitable numerical method for the approximation of discontinuous solutions of parameter-dependent linear hyperbolic conservation laws. The objective is to reconstruct such approximation, for new instances of the parameter values for any time, from a transformation of pre-computed snapshots of the solution trajectories for new parameter values. In a finite volume setting, a Reconstruct-Transform-Average (RTA) algorithm inspired from the Reconstruct-Evolve-Average one of Godunov's method is proposed. It allows to perform, in three steps, a transformation of the snapshots with piecewise constant reconstruction. The method is fully detailed and analyzed for solving a parameter-dependent transport equation for which the spatial transformation is related to the characteristic intrinsic to the problem. Numerical results for transport equation and linear elastodynamics equations illustrate the good behavior of the proposed approach.