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A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

Pei Fu, Gunilla Kreiss, Sara Zahedi

TL;DR

The paper develops high-order bound-preserving Cut-DG methods for one-dimensional hyperbolic conservation laws, allowing interior interfaces to cut through a regular background mesh. A novel macro-element reconstruction, combined with ghost-penalty stabilization, preserves conservation and convergence while enabling maximum principle and positivity preservation for scalar and Euler equations; high-order limiters are applied on macro-elements to control oscillations near shocks. Theoretical results establish maximum principle preservation for scalars and positivity for density and pressure in the Euler system under suitable time-step restrictions, with mean-value analyses supporting bound preservation for higher-order schemes. Numerical experiments demonstrate accuracy, bound preservation, and robust shock-capturing on cut meshes, including challenging problems such as Sedov blasts and multiple Riemann problems. The methodology promises effective extension to higher dimensions and complex interfaces, providing a practical unfitted DG framework with rigorous bound-preserving properties.

Abstract

In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and order of convergence. Our lowest-order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high-order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations and show that our scheme is positivity preserving. In the presence of shocks, additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

TL;DR

The paper develops high-order bound-preserving Cut-DG methods for one-dimensional hyperbolic conservation laws, allowing interior interfaces to cut through a regular background mesh. A novel macro-element reconstruction, combined with ghost-penalty stabilization, preserves conservation and convergence while enabling maximum principle and positivity preservation for scalar and Euler equations; high-order limiters are applied on macro-elements to control oscillations near shocks. Theoretical results establish maximum principle preservation for scalars and positivity for density and pressure in the Euler system under suitable time-step restrictions, with mean-value analyses supporting bound preservation for higher-order schemes. Numerical experiments demonstrate accuracy, bound preservation, and robust shock-capturing on cut meshes, including challenging problems such as Sedov blasts and multiple Riemann problems. The methodology promises effective extension to higher dimensions and complex interfaces, providing a practical unfitted DG framework with rigorous bound-preserving properties.

Abstract

In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and order of convergence. Our lowest-order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high-order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations and show that our scheme is positivity preserving. In the presence of shocks, additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.
Paper Structure (30 sections, 4 theorems, 69 equations, 14 figures, 2 algorithms)

This paper contains 30 sections, 4 theorems, 69 equations, 14 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A cut discontinuous Galerkin discretization in space
  3. A model geometry in one space dimension
  4. Mesh and spaces
  5. Macro-elements
  6. The weak formulation
  7. A reconstruction on macro-elements
  8. Maximum principle preserving DG methods for scalar equations
  9. Piecewise constant approximation
  10. Mean values of high order approximations
  11. A bound preserving limiter for scalar problems
  12. A Positivity preserving Cut-DG method for the Euler equations
  13. Piecewise constant approximations
  14. Mean values for higher order approximations
  15. Positivity preserving limiting for the Euler equations
  16. ...and 15 more sections

Key Result

Theorem 3.1

Consider the Cut-DG scheme described by scheme:cutDG2fully-eq:schemestep3 with $p=0$. If the time step $\Delta t$ satisfies then $u_{h,i}^n|_{\Omega_i} \in [\mathtt{m},\mathtt{M}],i=1,\cdots, N_\Omega$ implies $u^{n+1}_{h,i}|_{\Omega_i}\in[\mathtt{m},\mathtt{M}]$ for $i=1,\cdots, N_\Omega$. Here $h$ is the element size in the background mesh, and $\delta$ is a given constant, which in eq:largeel

Figures (14)

  • Figure 1: An illustration of a regular mesh and an unfitted interface.
  • Figure 2: Discretization of the computational domain $\Omega_h=[x_L,x_R]$ into a uniform partition. This background mesh is cut by two interfaces at $x=\Gamma_1$ and $x=\Gamma_2$, dividing the computational domain into three subdomains. The dotted line at $x_{j-\frac{3}{2}}$ indicates the interior edge in $\mathcal{M}_{h,1}$ on which stabilization is applied, i.e. $x_{j-\frac{3}{2}}\in\mathcal{F}_{h,1}$.
  • Figure 3: Illustration of intervals $K_j\in \mathcal{K}_{h,i}$, which constitute part of the macro-element M.
  • Figure 4: The errors in the numerical solutions for the advection equation with smooth initial data at $t=1.0$. Errors are shown in both the $L^2$-norm (solid lines) and $L^\infty$-norm (dashed lines). Left: time discretization using third order Runge-Kutta method. Right: time discretization using third order multi-step discretization in \ref{['eq:multi-step3']}.
  • Figure 5: Numerical $P^1$ (top), $P^2$ (bottom) solutions for the advection equation with non-smooth initial data at $t=1.0$. Left: without reconstruction and without limiter. Middle: without reconstruction but limiter \ref{['eq:limit:uj']} is applied. Right: solution with reconstruction and limiter \ref{['eq:limit:uj']} applied on macro-elements.
  • ...and 9 more figures

Theorems & Definitions (8)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof