Termination of Newton/Chord Iterations and the Method of Lines

Many ordinary diierential equation and diierential algebraic equation codes terminate the nonlinear iteration for the corrector equation when the diierence between successive iterates (the step) is suuciently small. This termination criterion avoids the expense of evaluating the non-linear residual at the nal 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 justiied 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 In this paper, we show how an out-of-date Jacobian coupled with moderate ill-conditioning can lead to premature termination of the corrector iteration and suggest ways in which this situation can be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 solution of Richards' equation, which models ow 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 eecient 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.