A non‐overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two‐dimension

Abstract

We apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two‐dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non‐overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.