A Simplified h-box Method for Embedded Boundary Grids

Abstract

We present a simplified $h$-box method for integrating time-dependent conservation laws on embedded boundary grids using an explicit finite volume scheme. By using a method of lines approach with a strong stability preserving Runge-Kutta method in time, the complexity of our previously introduced $h$-box method is greatly reduced. A stable, accurate, and conservative approximation is obtained by constructing a finite volume method where the numerical fluxes satisfy a certain cancellation property. For a model problem in one space dimension using appropriate limiting strategies, the resulting method is shown to be total variation diminishing. In two space dimensions, stability is maintained by using rotated $h$-boxes as introduced in previous work [M. J. Berger and R. J. LeVeque, Comput. Systems Engrg., 1 (1990), pp. 305-311; C. Helzel, M. J. Berger, and R. J. LeVeque, SIAM J. Sci. Comput., 26 (2005), pp. 785-809], but in the new formulation, $h$-box gradients are taken solely from the underlying Cartesian grid, which also reduces the computational cost.