Implementing a fully implicit numerical scheme in a 2-D fluid code with the Krylov method

In order to obtain a robust 2-D fluid code for modeling the edge-plasma region of a tokamak, the authors have implemented an implicit method-of-lines scheme utilizing the Krylov technique. The motivation for this work comes from the success of a different implicit scheme reported recently, and from the expectation that the electrostatic potential can be solved for unambiguously using a fully implicit algorithm. Here, the authors solve a coupled set of temporal ordinary differential equations for the variables (particle, parallel momentum, and electron and ion energy densities) at each spatial grid point. Portions of the existing B2 code are used to evaluate the spatial derivatives for the right-hand sides of the ODE's. The Krylov method which solves the ODE's does not require formation or solution of the Jacobian matrix explicitly. However, in order to obtain efficient performance, one needs to precondition the problem by obtaining an approximate solution. The potential payoff is that the required approximate solution may be cheaper to compute than the full Jacobian. The authors have used a variant of the original B2 solution as a preconditioner and also a banded matrix approximation to the Jacobian which keeps many of the couplings in the poloidal direction. Both preconditioners show a similar substantial increase in speed compared to using no preconditioner. The overall performance is not yet optimal in comparison to other diffusion problems which have used this technique. They report on additional modifications to the preconditioner in order to realize better performance, and compare the method to the original B2 solution in terms of speed and robustness. 3 refs