Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mukthesh Mahadev, Marc Gerritsma

TL;DR

The paper develops a high-order, structure-preserving space-time discretization for inviscid barotropic flow grounded in a multisymplectic variational principle. By operating on a fixed reference configuration and employing a mimetic spectral element method with primal–dual staggering, it achieves exact discrete conservation of mass, linear and angular momentum, and total energy, while naturally handling low-Mach regimes. The approach yields a fully discrete weak formulation that preserves the geometric structure of the continuous problem and avoids mesh distortion typical of traditional Lagrangian methods. Numerical experiments on expansion and compression tests confirm exact conservation to machine precision, stability, and convergence with increasing polynomial order, highlighting the method’s potential for robust long-time simulations of compressible-like barotropic flows.

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.

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

TL;DR

The paper develops a high-order, structure-preserving space-time discretization for inviscid barotropic flow grounded in a multisymplectic variational principle. By operating on a fixed reference configuration and employing a mimetic spectral element method with primal–dual staggering, it achieves exact discrete conservation of mass, linear and angular momentum, and total energy, while naturally handling low-Mach regimes. The approach yields a fully discrete weak formulation that preserves the geometric structure of the continuous problem and avoids mesh distortion typical of traditional Lagrangian methods. Numerical experiments on expansion and compression tests confirm exact conservation to machine precision, stability, and convergence with increasing polynomial order, highlighting the method’s potential for robust long-time simulations of compressible-like barotropic flows.

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.

Paper Structure

This paper contains 40 sections, 50 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Lagrangian Formulation of Barotropic Flow
  3. Hamilton's Principle and the Variational Formulation
  4. Mimetic Discretization
  5. Nodal (Lagrange) Basis.
  6. Edge Basis.
  7. Incidence matrices
  8. Dual Basis Functions
  9. 1D Dual Basis Functions
  10. Variable Representation in Two Dimensional Space and Time $\mathbb{R}^{2+1}$
  11. Primal-Dual Variable Staggering
  12. Configuration (Kinematic) Variables:
  13. Source (Dynamic) Variables:
  14. Spectral Basis Expansion
  15. Primal Nodal Basis:
  16. ...and 25 more sections

Figures (16)

  • Figure 1: Reference and Deformed Configurations with Flow Map $\varphi(t, X)$
  • Figure 2: 1D basis functions for polynomial degree N = 4
  • Figure 3: 1D Dual basis functions for polynomial degree N = 4
  • Figure 4: Localization of degrees of freedom on the primal mesh.
  • Figure 5: First Piola-Kirchhoff stress degrees of freedom on dual spatio-temporal faces.
  • ...and 11 more figures

Theorems & Definitions (3)

  • proof : Proof
  • proof : Proof
  • proof : Proof