The use of block tridiagonal methods to improve convergence of DADI solutions to coupled elliptic partial differential equations

Abstract

A new solution method for coupled sets of elliptic equations, Coupled Equation Dynamic Alternating Direction Implicit (CEDADI), was recently developed for solution of the Streamlined Darwin Field (SDF) model and shows promise for other applications. In the SDF simulations of plasmas, CEDADI has been shown to dramatically reduce the required CPU time to achieve a chosen residual. In these simulations, the coupling between the equations was very strong. For intermediate equation coupling which will occur in general sets of coupled elliptic equations, the method slows considerably. We will show that the problem is corrected by time centering both second and first-order derivatives. The iteration count is then reduced by a factor of 8 for the intermediate coupling case. The new time centering requires a block tridiagonal inversion in each direction instead of the previously used tridiagonal inversion. Despite the fact that the new method takes more CPU time per iteration compared to the old method, an overall factor of 6.28 reduction in CPU time is then gained for this case. Reduction in CPU time is obtained for cases over a wide range of coupling parameters.