Development of discontinuous Galerkin methods for hyperbolic systems that preserve a curl or a divergence constraint: the case of linear systems

Abstract

Some hyperbolic systems are known to include implicit preservation of differential constraints: these are for example the time conservation of the curl or the divergence of a vector that appear as an implicit constraint. In this article, we show that this kind of constraint can be easily conserved at the discrete level with the classical discontinuous Galerkin method, provided the right approximation space is used for the vectorial space, and under some mild assumption on the numerical flux. For this, we recall a discrete de-Rham framework in which discontinuous approximation spaces for vectors fits. The discrete adjoint divergence and curl are proven to be exactly preserved by the discontinuous Galerkin method under a small assumption on the numerical flux. Numerical tests are performed on the wave system, the two dimensional Maxwell system and the induction equation, and confirm that the differential constraints are preserved at machine precision while keeping the high order of accuracy.

Figures (10)

• Figure 1: The meshes on which the conservation of divergence or curl are performed. Top left: Cartesian mesh, top right: unstructured quadrangular mesh, bottom: unstructured triangular mesh.
• Figure 2: Plot of $\Arrowvert \nabla^\star \mathbf{e} - \nabla^\star \mathbf{e}_0 \Arrowvert_2$ with respect to time for different degree, approximation space, type of meshes and numerical flux. In the left column, Cartesian and unstructured quadrangular meshes are considered, with the $\pmb{\mathrm{d}\mathbb{Q}}_k$ approximation space with the Godunov (top figure) and Lax-Friedrich (bottom figure) numerical flux. In the right column, Cartesian, unstructured quadrangular and triangular meshes are considered, with the $\pmb{\mathrm{d}\mathbb{B}}^{\mathrm{curl}}_k$ approximation space with the Godunov (top figure) and Lax-Friedrich (bottom figure) numerical flux. On triangular meshes, the space $\pmb{\mathrm{d}\mathbb{B}}^{\mathrm{curl}}_k$ and $\pmb{\mathrm{d}\mathbb{P}}_k$ are the same, and the computations on these spaces are represented on the right column. Note that the $y$ scaling of the top right figure ($\pmb{\mathrm{d}\mathbb{B}}^{\mathrm{curl}}_k$ with Godunov' scheme) is $10^{-12}$, whereas it is of the order of $1$ for the other plots.
• Figure 3: Error obtained on the test case described in \ref{['subsubsec:ConvergenceDivergence']} with initial condition \ref{['eq:exactDivergence']} on the left, and with the initial condition \ref{['eq:DivergenceAddVortex']} added to \ref{['eq:exactDivergence']} on the right, on a series of Cartesian meshes. For each of the test cases, the error obtained for the variables $b$ (top row), $\mathbf{e}_x$ (middle row) and $\mathbf{e}_y$ (bottom row) is shown for different approximation spaces and for the Lax-Friedrich and Godunov flux.
• Figure 4: Error obtained on the test case described in \ref{['subsubsec:ConvergenceDivergence']} with initial condition \ref{['eq:exactDivergence']} on the left, and with the initial condition \ref{['eq:DivergenceAddVortex']} added to \ref{['eq:exactDivergence']} on the right, on a series of unstructured triangular meshes. For each of the test cases, the error obtained for the variables $b$ (top row), $\mathbf{e}_x$ (middle row) and $\mathbf{e}_y$ (bottom row) is shown for different degrees, and for the Lax-Friedrich and Godunov numerical flux.
• Figure 5: Plot of $\Arrowvert \left( {\nabla^{\perp}} \right)^\star \mathbf{u} - \left( {\nabla^{\perp}} \right) ^\star \mathbf{u}_0 \Arrowvert_2$ with respect to time for different degree, approximation space, type of meshes and numerical flux. In the left column, Cartesian and unstructured quadrangular meshes are considered, with the $\pmb{\mathrm{d}\mathbb{Q}}_k$ approximation space with the Godunov (top figure) and Lax-Friedrich (bottom figure) numerical flux. In the right column, Cartesian, unstructured quadrangular and triangular meshes are considered, with the $\pmb{\mathrm{d}\mathbb{B}}^{\mathrm{div}}_k$ approximation space with the Godunov (top figure) and Lax-Friedrich (bottom figure) numerical flux. On triangular meshes, the space $\pmb{\mathrm{d}\mathbb{B}}^{\mathrm{div}}_k$ and $\pmb{\mathrm{d}\mathbb{P}}_k$ are the same, and the computations on these spaces are represented only on the right column. Note that the $y$ scaling of the top right figure ($\pmb{\mathrm{d}\mathbb{B}}^{\mathrm{div}}_k$ with Godunov' scheme) is $10^{-12}$, whereas it is of the order of $1$ for the other plots.

Theorems & Definitions (19)

• Definition 1: Adjoint differential operators
• Definition 2: Projection operator on $\mathbb{C}_k$
• Proposition 1: Commutation property of the projection operator on $\mathbb{C}_k$
• proof
• Proposition 2: Conservation of $\left( {\nabla^{\perp}} \mathbf{u} \right)^{\star}$
• proof
• Remark 1: full discretization in space: effect of quadrature formula
• Remark 2: Time discretization
• Remark 3: Lax-Friedrich flux with normal diffusion
• Remark 4: Link with the Hodge Laplacian