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An arbitrarily high-order accurate and locally energy/entropy-stable space-time method for a class of nonlinear wave systems

Michael Dumbser, Ilaria Perugia, Enrico Zampa

arXiv:2610.09797v1math.NAcs.NA

TL;DR

Nonlinear wave systems may develop shocks, but viscous regularization must also preserve energy/entropy behavior and the curl involution. The paper introduces a DG space-time method using projected dual variables and reports local conservation, energy/entropy balance, and weak curl preservation, including with discontinuities.

  • Problem

    Nonlinear hyperbolic wave systems may develop shocks, while standard viscous regularization is numerically unattractive because most methods do not satisfy the chain rule.

    Source: §2.1 · pp. 5–6

  • Method

    The method represents state and dual variables in the same DG space, computes dual variables as L2 projections of energy derivatives, and uses a continuous-discontinuous Petrov–Galerkin method in time.

    Source: §6 · p. 19

  • Results

    The schemes satisfy local conservation and local energy/entropy balance while weakly preserving curl, and the circular-explosion test captures shocks with suppressed oscillations and dissipates energy/entropy.

    Source: §5.3 · p. 19, §6 · p. 20

  • Takeaways & Limitations

    The approach requires no special quadrature rules or specific basis functions and retains a standard, solution-independent mass matrix for explicit time integration.

    Source: §1 · p. 3

    The alternative stabilization does not preserve the involution and is recommended only in one dimension or when the involution is irrelevant.

    Source: §3.4.1 · p. 12

Abstract

from arXiv · show

We present a new class of structure-preserving, arbitrarily high-order accurate space-time schemes using unstructured simplicial meshes in space for a class of nonlinear acoustic wave propagation systems, in which the nonlinearity is encoded in the constitutive relations between the state and dual variables, while the differential operators retain the standard first-order acoustic wave structure. At the fully discrete level, the schemes satisfy the following properties: I) local conservation of the state variables; II) a local energy/entropy balance on each space-time element, with a consistent, single-valued numerical energy/entropy flux and nonnegative local energy/entropy dissipation, and exact energy conservation when the stabilization terms are omitted; III) weak preservation of the curl involution of the velocity field, even in the presence of discontinuities and with the shock-capturing terms activated. The key ingredients of the new schemes are: i) a discontinuous Galerkin (DG) discretization in space, in which the dual variables are computed as the L2 projections onto the DG space of the energy derivatives with respect to the conserved variables; ii) numerical fluxes and a bubble-weighted, element-local viscous regularization expressed in terms of the dual variables; iii) a continuous-discontinuous Petrov-Galerkin method in time. Properties I)-III) are first proved mathematically and are then verified by suitable numerical tests.

1 Introduction

Invariant-preserving methods often emphasize global structure, while local preservation is less common and matters for localized phenomena such as shocks. The paper proposes a DG and CG–DG approach for nonlinear acoustic waves that targets local invariants, high order, unstructured meshes, and dissipative shock capturing.

1 Introduction

  • Global conservation of a coercive invariant may not capture localized phenomena such as shocks, while local invariant-preserving methods remain relatively scarce.The related work also describes geometric and Hamiltonian approaches as typically focused on global structures.Source: §1 · p. 2
  • Entropy-stable DG approaches include change-of-variable formulations, SBP and specialized-flux frameworks, and HTC schemes, each with distinct implementation requirements.Change-of-variable methods can require nonlinear mass matrices and local nonlinear recovery; SBP frameworks require tailored quadrature and algebraic constructions; HTC methods evolve entropy density and obtain local total-energy conservation as a consequence.Source: §1 · p. 2
  • The proposed method combines conservative and dual variables to preserve relevant invariants locally and globally, with a CG–DG time discretization that makes the structural properties hold fully discretely.Energy is exactly conserved when stabilization and viscous regularization are omitted; explicit Runge–Kutta retains local conservation and the curl involution but not the exact energy/entropy balance.Source: §1 · p. 3
  • The nonlinear acoustic model confines nonlinearity to constitutive relations, and the paper pursues arbitrary order, unstructured spatial meshes, local energy balances, and dissipative shock capturing.This differs from the cited second-order, staggered-Cartesian approach [55].Source: §1 · p. 3

2 Governing equations

The paper studies nonlinear acoustic systems through their energy structure, then introduces a viscous regularization and a global Hamiltonian interpretation. These formulations motivate a method designed to dissipate energy locally while preserving the curl involution and other structural properties.

2 Governing equations

  • The governing model is a nonlinear acoustic wave system with convex energy; positive-definite energy Hessians make it symmetric hyperbolic, and its strong solutions satisfy a local energy conservation law.The construction extends to an abstract system that includes nonlinear Maxwell at k = 1, although the presentation is confined to the acoustic system.Source: §2 · pp. 4–5

2.1 Viscous regularization

  • The naive viscous regularization dissipates energy locally for convex energy, but is numerically unattractive because most methods do not satisfy its required chain rule.The alternative regularization is motivated by preserving the curl involution while dissipating energy/entropy.Source: §2.1 · pp. 5–6
  • With positive coefficient fields, the alternative regularization yields nonnegative local energy/entropy dissipation and preserves the curl involution.The nonnegative dissipation holds regardless of whether the energy is convex.Source: §2.1 · p. 6

2.2 Global Hamiltonian point of view

  • The regularized system has a Lie–Poisson form with a skew-symmetric operator and a negative semi-definite gradient-flow operator, giving a global energy/entropy balance.Prior discretizations in this perspective focus on global structures and often neglect local structural properties; this work also targets nonlinear Hamiltonians, involution preservation, and shock capture.Source: §2.2 · p. 7

3 Spatial semidiscretization

The semidiscrete DG method combines dual variables, numerical fluxes, and element-local viscous regularization to preserve local conservation, energy/entropy stability, and the curl involution. The section also gives a different stabilization strategy and a hybridized variant for quadratic energy with u = v.

3.1 Notation, mesh, forms and polynomial spaces

  • On a conforming simplicial mesh, element-local bubble-weighted forms regularize divergence of the velocity dual variable and gradients of the density dual variable.The regularization coefficient ε is intended to be near zero in smooth regions and O(h) near shocks or discontinuities.Source: §3.1 · pp. 7–8

3.2 The semidiscrete method

  • The spatial DG equations use average-based fluxes for consistency and jump terms for stabilization, with the wall condition imposed weakly through the numerical flux.The bubble-weighted forms avoid interface averages and an additional penalty parameter required by an interior-penalty alternative.Source: §3.2 · pp. 8–9

3.3 Properties

  • Each element satisfies local conservation laws for the velocity components and density, alongside a local energy/entropy balance with nonnegative dissipation and a consistent, single-valued flux.The dissipation vanishes and local energy is conserved when ε = βh = 0; summing element balances gives the global energy balance.Source: §3.3 · pp. 9–11
  • The DG method weakly preserves the curl involution of the velocity field.The involution is expressed variationally through orthogonality against curls of admissible test functions.Source: §3.3 · p. 11

3.4.1 A different stabilization strategy

  • That alternative stabilization does not preserve the involution, so it is recommended only in one dimension or when the involution is irrelevant.Source: §3.4.1 · p. 12
  • An alternative damping strategy remains entropy stable with every projection level retained because damping acts on dual variables that test the energy balance.Each projection level therefore contributes a nonnegative term to local energy/entropy dissipation.Source: §3.4.1 · p. 12

3.4.2 A variant for the case u = v

  • For quadratic energy with u = v, a div-conforming velocity space permits a variant without the bubble function, and hybridization recovers local conservation and the energy/entropy balance.The variant's numerical fluxes are single-valued and vanish on wall facets.Source: §3.4.2 · pp. 12–13

4 Fully discrete space–time method

The fully discrete method combines a space-time DG discretization with projected dual variables and a continuous-discontinuous Petrov-Galerkin time method. Its stated properties include local conservation, local energy/entropy balance with nonnegative dissipation, and vanishing dissipation when stabilization is omitted.

4 Fully discrete space–time method

  • A continuous-trial, discontinuous-test Petrov-Galerkin time discretization gives a fully implicit scheme whose space-time slabs couple elements through numerical fluxes.A global nonlinear system must be solved at each time step.Source: §4 · p. 14
  • The method projects the energy derivatives onto the space-time DG polynomial space for its dual variables, combining this projection with spatial DG to obtain a local discrete energy/entropy balance.The projection is related to established energy-preserving time discretizations; the stated novelty is its combination with DG in space.Source: §4 · p. 15
  • The fully discrete scheme satisfies a local energy/entropy balance on each space-time element, with dissipation vanishing when ε = βhx = 0.The theorem also states local conservation and global energy/entropy balance.Source: §4 · pp. 15–16

5 Numerical results

The numerical tests assess conservation, smooth-solution convergence, and shock behavior for conservative and dissipative variants. The reported results show machine-precision conservation, optimal convergence for both variables with stabilization, and effective shock capture with weak curl remaining near machine precision.

5 Numerical results

  • The scheme is not proven to preserve positive density, although density remains positive in the reported numerical experiments.Source: §5 · p. 16
  • The tests compare a conservative variant with β_h = ε = 0 and a dissipative variant with β_h = 1/2 and elementwise artificial viscosity.The viscosity acts as shock capturing and remains small enough for smooth-solution high-order convergence.Source: §5 · p. 17
  • Energy/entropy stability does not require a CFL condition, but the timestep must support nonlinear solver convergence and density positivity.The nonlinear systems are solved with Newton’s method and GMRES using a block-Jacobi preconditioner.Source: §5 · p. 17

5.1 Conservation properties

5.2 Convergence rates

  • The dissipative scheme achieves k+1 convergence for both ρ and v, while the conservative scheme achieves k+1 for ρ and k for v.The reported smooth-solution test finds that stabilization improves v’s convergence rate without spoiling the optimal rate.Source: §5.2 · p. 19, Table 2 · p. 19

5.3 Circular explosion

  • At t=0.3, the dissipative scheme captures the shock with oscillations essentially suppressed, dissipates global energy/entropy, and keeps weak curl near machine precision.The density is compared with a fine-mesh MUSCL–Hancock reference solution.Source: §5.3 · p. 19

6 Conclusions

The paper introduces high-order space–time schemes for nonlinear wave systems, combining DG in space with a continuous–discontinuous Petrov–Galerkin method in time. The schemes are locally conservative, satisfy local energy/entropy balances, and weakly preserve the velocity curl involution, including in the presence of discontinuities.

6 Conclusions

  • The method combines a DG spatial discretization on unstructured meshes with a continuous–discontinuous Petrov–Galerkin discretization in time.Source: §6 · p. 19
  • State and dual variables share a DG space, with dual variables obtained as L2 projections of energy derivatives with respect to conserved variables.The dual variables are subsequently used as test functions in the energy/entropy balance.Source: §6 · pp. 19–20
  • Figure 3 presents the two-dimensional circular explosion problem.Source: Figure 3 · p. 20
  • The schemes are locally conservative, satisfy local energy/entropy balances, and weakly preserve the velocity curl involution, including with discontinuities when viscous regularization is used.Source: §6 · p. 20
  • Future work will address nonlinear Maxwell systems and more complex hyperbolic systems with fluxes that are nonlinear functions of dual variables, including compressible Euler equations.Source: §6 · p. 20
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