A Mixed Explicit Implicit Time Stepping Scheme for Cartesian Embedded Boundary Meshes

We present a mixed explicit implicit time stepping scheme for solving the linear advection equationMay, Sandra on a Cartesian embedded boundary mesh. The scheme represents a new approach for overcoming the small cell problem-that standard finite volume schemes are not stable on the arbitrarily small cut cells. It uses implicit time stepping on cut cells for stability. On standard Cartesian cells, explicit time stepping is employed. This keeps the cost small and makes it possible to extend existing schemes from Cartesian meshes to Cartesian embedded boundary meshes. The coupling is done by flux bounding, for which we can prove a TVD result. We present numerical results in one and two dimensions showing second-order convergence in the L-1 norm and between first- and second- order convergence in the L-infinity norm.