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Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections

Elena Gaburro, Walter Boscheri, Simone Chiocchetti, Mario Ricchiuto

TL;DR

This work tackles the challenge of solving hyperbolic systems in non-conservative variables while accurately capturing shocks. It develops a high-order ADER-DG method that evolves non-conservative primitive variables but enforces local conservation in shocks via an a posteriori subcell FV correction, forming a quasi-conservative framework. A vanishing conservation defect is defined and analyzed, enabling a modified Lax-Wendroff theorem; shocks are treated with a shock-detecting limiter that switches to a locally conservative FV update in troubled cells. The approach yields high-order accuracy in smooth regions, robust non-oscillatory shock resolution, and reliable multi-material interface handling, with numerical results on Euler and multi-material Euler benchmarks demonstrating precise shock speeds, accurate contacts, and suppression of spurious oscillations. The method offers a practical path to efficiently simulate complex multi-material flows directly in physically meaningful variables, while preserving formal convergence properties where the conservation defect vanishes.

Abstract

We present a novel quasi-conservative arbitrary high order accurate ADER discontinuous Galerkin (DG) method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax-Wendroff theorem. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax-Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics. To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble.

Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections

TL;DR

This work tackles the challenge of solving hyperbolic systems in non-conservative variables while accurately capturing shocks. It develops a high-order ADER-DG method that evolves non-conservative primitive variables but enforces local conservation in shocks via an a posteriori subcell FV correction, forming a quasi-conservative framework. A vanishing conservation defect is defined and analyzed, enabling a modified Lax-Wendroff theorem; shocks are treated with a shock-detecting limiter that switches to a locally conservative FV update in troubled cells. The approach yields high-order accuracy in smooth regions, robust non-oscillatory shock resolution, and reliable multi-material interface handling, with numerical results on Euler and multi-material Euler benchmarks demonstrating precise shock speeds, accurate contacts, and suppression of spurious oscillations. The method offers a practical path to efficiently simulate complex multi-material flows directly in physically meaningful variables, while preserving formal convergence properties where the conservation defect vanishes.

Abstract

We present a novel quasi-conservative arbitrary high order accurate ADER discontinuous Galerkin (DG) method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax-Wendroff theorem. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax-Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics. To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble.
Paper Structure (40 sections, 5 theorems, 120 equations, 10 figures, 1 table)

This paper contains 40 sections, 5 theorems, 120 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Multi-material compressible Euler equations
  4. ADER-DG framework for hyperbolic systems in non-conservative form
  5. Geometry and notation
  6. Remark
  7. Local space-time ADER predictor in physical variables
  8. Path-integrated discontinuous Galerkin corrector
  9. A posteriori subcell finite volume limiter
  10. Subcell definition and averaging
  11. Troubled cell detection
  12. Subcell finite volume update
  13. DG polynomial reconstruction
  14. Quasi-conservative formulation via a posteriori local conservative subcell corrections
  15. Local conservation, Lax-Wendroff theorem, and vanishing conservation defect
  16. ...and 25 more sections

Key Result

Proposition 1

Consider a scheme whose discrete solution $\mathbf{q}_h$ verifies the modified local conservation statement eqn.localQuasiConservation0. Assume that the discrete initial condition $\mathbf{q}_h^0$ verifies the weak convergence estimate and that the discrete solution also verifies, in a ball $B_{i}$ centered around the center of the cell $\omega_i$, the total-variation-boundedness condition in the

Figures (10)

  • Figure 1: Numerical results obtained with our third $P_2$ (left) and fourth $P_3$ (right) order scheme for the planar shock wave propagation with Mach $M=5$ (dark-green), $M=10$ (blue) and $M=20$ (red). We also report with dashed black lines the expected position of each shock wave at the final time $t_f=0.04$ to show that our scheme perfectly deals with shock waves modeling.
  • Figure 2: Lax shock tube solved on a polygonal tessellation with characteristic mesh size $h=1/430$. We report the results obtained with our quasi-conservative third order $P_2$ (top) and fourth order $P_3$ (bottom) schemes. In particular, we show on the left a graph whose $z-$coordinate is given by the density and whose colors refer to the type of scheme employed on each cell: blue for our quasi-conservative scheme, green for the second order limiter in primitive variables and red for the second order limiter applied on the conservative formulation, which is used to discretize the solution where the shock wave is located. Then, on the right, we report a scatter plot of our numerical results for the density profile compared with a reference solution obtained with a one dimensional Euler solver using a fully conservative explicit second order limited Residual Distribution scheme (see e.g. RICCHIUTO20105653AR:17) on 50000 points.
  • Figure 3: Circular Sod explosion. We show the numerical results obtained with our quasi-conservative third order $P_2$ (top row) and fourth order $P_3$ (bottom row) discontinuous Galerkin schemes. On the left, we depict in blue the cells where our DG scheme has been applied to the non-conservative formulation and in red or green those where our second order subcell FV limiter has been activated. In particular, on the red cells which correspond to shock-triggered troubled cells the FV schemes is applied on the conservative formulation of the Euler equations. On the right, we report the scatter plot of the numerical solution obtained on a coarse mesh $M_1$ with characteristic mesh size of $h=1/50$ and a finer one $M_2$ with characteristic mesh size of $h=1/215$, compared with a reference solution (black), and the incorrect solution (green) that one could have obtained by applying the DG scheme for the non-conservative formulation everywhere on the domain, including the shock regions.
  • Figure 4: Woodward-Colella blast waves at the final time $t= 0.038$ solved on a polygonal tessellation with characteristic mesh size $h=1/1080$. We report the results obtained with our third order $P_2$ (top row) and fourth order $P_3$ (bottom row) discontinuous Galerkin schemes, which solve the Euler equations formulated in primitive variables everywhere a part for the shock regions, highlighted in red on the left, where a second order FV scheme has been applied to the conservative formulation of the PDE at the subgrid level. In particular, on the right we report a zoom on the region $x\in[0.5,1]$ where we compare the scatter plot of our numerical results with a reference solution obtained with a fully conservative non-linear Residual Distribution scheme RICCHIUTO20105653AR:17 applied to the 1D Euler equations on 50000 points.
  • Figure 5: Numerical results for the forward facing step benchmark at time $t=2.5$ and $t=4$ obtained with our quasi-concervative third order $P_2$ (first and third image) and fourth order $P_3$ (second and fourth image) DG schemes on a polygonal tessellation with characteristic mesh size of $h=1/80$.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Proposition 1: Local quasi-conservation condition and Lax-Wedroff theorem
  • proof
  • Proposition 2: Conservation defect: locally regular solutions
  • proof
  • Proposition 3: Conservation defect: regular solutions and subcell non-conservative update
  • proof
  • Proposition 4: Conservation defect: data mismatch on cell boundaries
  • Proposition 5: Contact discontinuities and mass/momentum/energy conservation defect
  • proof