Stability analysis of discontinuous Galerkin with a high order embedded boundary treatment for linear hyperbolic equations

TL;DR

The paper analyzes the stability of high-order discontinuous Galerkin methods coupled with the shifted boundary polynomial correction for linear hyperbolic equations in 1D. By examining the eigenvalue spectrum of a simplified two-element DG system, it shows that explicit time integration imposes nontrivial stability restrictions when the boundary is immersed at a distance $d=\mathcal{O}(\Delta x)$ from the mesh, with asymmetry between exterior and interior boundary placements. Implicit time integration greatly relaxes, and can remove, these restrictions, yielding unconditional stability for suitable CFL values; numerical experiments corroborate the theoretical predictions across polynomial degrees $p=1,2,3$. The findings inform the practical use of high-order embedded boundary treatments in hyperbolic problems and motivate extensions to multi-dimensional settings and alternative discretizations.

Abstract

Embedded, or immersed, approaches have the goal of reducing to the minimum the computational costs associated with the generation of body-fitted meshes by only employing fixed, possibly Cartesian, meshes over which complex boundaries can move freely. However, this boundary treatment introduces a geometrical error of the order of the mesh size that, if not treated properly, can spoil the global accuracy of a high order discretization, herein based on discontinuous Galerkin. The shifted boundary polynomial correction was proposed as a simplified version of the shifted boundary method, which is an embedded boundary treatment based on Taylor expansions to deal with unfitted boundaries. It is used to accordingly correct the boundary conditions imposed on a non-meshed boundary to compensate the aforementioned geometrical error, and reach high order accuracy. In this paper, the stability analysis of discontinuous Galerkin methods coupled with the shifted boundary polynomial correction is conducted in depth for the linear advection equation, by visualizing the eigenvalue spectrum of the high order discretized operators. The analysis considers a simplified one-dimensional setting by varying the degree of the polynomials and the distance between the real boundary and the closest mesh interface. The main result of the analysis shows that the considered high order embedded boundary treatment introduces a limitation to the stability region of high order discontinuous Galerkin methods with explicit time integration, which becomes more and more important when using higher order methods. The implicit time integration is also studied, showing that the implicit treatment of the boundary condition allows one to overcome such limitation and achieve an unconditionally stable high order embedded boundary treatment.

Figures (11)

• Figure 1: These numerical results, obtained using a third order accurate $\mathbb{P}^2$ DG approximation of problem \ref{['eq:linear_advection_intro']}, show the impact of the distance on the extrapolation of Dirichlet boundary condition using the shifted boundary embedded approach. On the left, the distance $d$ between the real boundary and the mesh is taken equal to $h^2$, mimicking a curved boundary discretized with simplicial elements, and the solution stays stable. On the right, $d$ is taken equal to $h/2$, mimicking a fully non-meshed configuration, and the solution is unstable.
• Figure 2: Mesh configuration: tessellation of one-dimensional elements with a boundary condition (red circle) imposed at a distance $d$ from the left-most interface of the domain, which is considered as the surrogate boundary of the problem.
• Figure 3: Boundary position: the real boundary position can be either outside of the first mesh element (above), or inside the first mesh element (below). In the first case, the distance will be taken as negative $d<0$, while for the second one the distance is taken positive $d>0$.
• Figure 4: Periodic boundary configuration used in the analysis of section \ref{['sec:periodic']}.
• Figure 5: Amplification factor: periodic boundary conditions for the explicit discontinuous Galerkin method.