A Cartesian Grid Embedded Boundary Method for the Heat Equation on Irregular Domains

We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (1998, J. Comput. Phys.147, 60) for discretizing Poissonʹs equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case in which the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank–Nicolson time discretization is unstable, requiring us to use the L0-stable implicit Runge–Kutta method of Twizell, Gumel, and Arigu (1996, Adv. Comput. Math.6, 333).