ACM Transactions on Mathematical Software (TOMS)
ADI schemes for higher-order nonlinear diffusion equations
Applied Numerical Mathematics
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Many ordinary differential equation (ODE) and differential algebraic equation (DAE) codes terminate the nonlinear iteration for the corrector equation when the difference between successive iterates (the step) is sufficiently small. This termination criterion avoids the expense of evaluating the nonlinear residual at the final iterate. Similarly, Jacobian information is not usually computed at every time step but only when certain tests indicate that the cost of a new Jacobian is justified by the improved performance in the nonlinear iteration. In this paper, we show how an out-of-date Jacobian coupled with moderate ill conditioning can lead to premature termination of the corrector iteration and suggest ways in which this situation can be detected and remedied. As an example, we consider the method of lines (MOL) solution of Richards' equation (RE), which models flow through variably saturated porous media. When the solution to this problem has a sharp moving front, and the Jacobian is even slightly ill conditioned, the corrector iteration used in many integrators can terminate prematurely, leading to incorrect results. While this problem can be solved by tightening the tolerances for the solvers used in the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver and/or recompute the Jacobian more frequently. Of these two, recomputation of the Jacobian is the more important. We propose a criterion based on an estimate of the norm of the time derivative of the Jacobian for recomputation of the Jacobian and a second criterion based on a condition estimate for tightening of the termination criteria of the nonlinear solver.