Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 42, Issue 6, Pages A3677-A3703Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1268318
Keywords
cut cell; unfitted finite elements; discontinuous Galerkin method; stabilization; small cell problem; hyperbolic conservation law
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We present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous Galerkin (DG) method in two dimensions with piecewise linear polynomials. The goal is to allow for explicit time stepping schemes despite the presence of cut cells. Using a method of lines approach, we start with a standard upwind DG discretization for the background mesh and add penalty terms that stabilize the solution on small cut cells in a conservative way. Then one can use explicit time stepping, even on cut cells, with a time step length that is appropriate for the background mesh. In one dimension, we show monotonicity of the proposed scheme with a constant basis and total variation diminishing in the means stability for piecewise linear polynomials. We also present numerical results in one and two dimensions that support our theoretical findings.
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