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-dep endent domains. lt is based on the Cartesian grid embedded boundary algorit hm of Johansen and Colella (1998, J. Comput. Phys. 147, 60) for discretizin g Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurat e in space and time. For the case in which the boundary is moving, we conve rt the moving-boundary problem to a sequence of fixed-boundary problems, co mbined with an extrapolation procedure to initialize values that are uncove red as the boundary moves. We find that, in the moving boundary case, the u se of Crank-Nicolson time discretization is unstable, requiring us to use t he Lo-stable implicit Runge-Kutta method of Twizell, Gumel, and Arigu (1996 , Adv. Comput. Math. 6,333). (C) 2001 Academic Press.