Space-time discretization for barotropic flow stemming from a multisymplectic variational formulation

TL;DR

This paper introduces a novel, structure-preserving space-time discretization method for inviscid barotropic flows based on a multisymplectic variational principle, ensuring exact conservation laws and stability.

Contribution

It develops a higher-order, multisymplectic variational discretization on a fixed reference configuration, improving stability and conservation in Lagrangian flow simulations.

Findings

Accurate and stable numerical results for expansion and compression flows.

Exact conservation of mass, momentum, and energy up to machine precision.

Handles low-Mach flows without special preconditioning.

Abstract

This study proposes and analyses a novel higher-order, structure preserving discretization method for inviscid barotropic flows from a Lagrangian perspective. The method is built on a multisymplectic variational principle discretized over a full space-time domain. Flow variables are encoded on a staggered space-time mesh, leveraging the principles of mimetic spectral element discretization. Unlike standard Lagrangian methods, which are prone to mesh distortion, this framework computes fluid deformations in a fixed reference configuration and systematically maps them to the physical domain via the Piola-Kirchhoff stress. Further, the structure preserving design ensures that the discrete analogues of the fundamental conservation laws for mass, momentum, and energy are satisfied up to machine precision. The formulation also inherently handles low-Mach number flows without specialized preconditioning. Numerical experiments on expansion and compression flows confirm the accuracy, stability, and exact conservation properties of the discretization.