Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Optimum positive linear schemes for advection in two and three dimensions
SIAM Journal on Numerical Analysis
Use of a rotated Riemann solver for the two-dimensional Euler equations
Journal of Computational Physics
An accuracy assessment of Cartesian-mesh approaches for the Euler equations
Journal of Computational Physics
An adaptive Cartesian grid method for unsteady compressible flow in irregular regions
Journal of Computational Physics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
A two-dimensional conservation laws scheme for compressible flows with moving boundaries
Journal of Computational Physics
A higher-order boundary treatment for Cartesian-Grid method
Journal of Computational Physics
H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids
SIAM Journal on Numerical Analysis
A High-Resolution Rotated Grid Method for Conservation Laws with Embedded Geometries
SIAM Journal on Scientific Computing
A second order kinetic scheme for gas dynamics on arbitrary grids
Journal of Computational Physics
A Cartesian grid embedded boundary method for hyperbolic conservation laws
Journal of Computational Physics
A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations
Journal of Scientific Computing
Partially implicit peer methods for the compressible Euler equations
Journal of Computational Physics
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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.