An efficient cell-centered diffusion scheme for quadrilateral grids

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: a b s t r a c t A new algorithm for solution of diffusion equations in two dimensions on structured quad-rilateral grids is proposed. The algorithm is based on a semi-implicit method for the time discretization and has a nine-point local stencil in space. Our scheme is fast, quite accurate and demonstrates good spatial convergence. The presented numerical tests show that it is well suited for hydrocodes with cell-centered principal variables. In this paper we describe a numerical algorithm for solution of the diffusion equation in two dimensions on quadrilateral grids, which appears to provide an excellent compromise between simplicity of realization, efficiency (in terms of the required computer time) and accuracy. Efficiency becomes a key issue if the diffusion solver is to be incorporated into a two-dimensional (2D) or three-dimensional (3D) hydrodynamics code. From general considerations it is clear that major improvements in efficiency for multi-dimensional schemes can be achieved if one chooses (i) a semi-implicit rather than fully implicit approach with respect to time differencing and (ii) a linear spatial differencing scheme. As had been proven by Kershaw [1], a linear second-order scheme on an arbitrary 2D mesh cannot be monotone. Monotonicity can be restored with non-linear algorithms [2,3], but such algorithms entail iterative solution of a large system of non-linear equations and are rather costly. Aiming at high performance efficiency, we stay by linear approach and compare our algorithm with the best earlier published schemes from this class [4–6]. Departures from monotonicity do not appear to be a serious issue in practical applications. Ideally, one would prefer to use a first-order fully implicit time discretization, which is robust, unconditionally stable and sets no additional time-step limit. However, solution of the corresponding large system of linear equations becomes a real challenge for certain versions of spatial discretization [5], and is usually rather costly in two and three dimensions. Here, a major simplification can be achieved by employing a symmetric semi-implicit (SSI) method proposed by Livne and Glasner [7], which is easy to implement and quite efficient in terms of megaflops per time step. But the decisive advantage of this method emerges when one tries to incorporate even …