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. lt is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (1998, J. Comput. Phys. 147, 60) for discretizing Poisson's 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 Lo-stable implicit Runge-Kutta method of Twizell, Gumel, and Arigu (1996, Adv. Comput. Math. 6,333). (C) 2001 Academic Press.