TERMINATION OF NEWTON CHORD ITERATIONS AND THE METHOD OF LINES/

Many ordinary differential equation (ODE) and differential algebraic e quation (DAE) codes terminate the nonlinear iteration for the correcto r equation when the difference between successive iterates (the step) is sufficiently small. This termination criterion avoids the expense o f evaluating the nonlinear residual at the final iterate. Similarly, J acobian information is not usually computed at every time step but onl y when certain tests indicate that the cost of a new Jacobian is justi fied by the improved performance in the nonlinear iteration. In this p aper, we show how an out-of-date Jacobian coupled with moderate ill co nditioning can lead to premature termination of the corrector iteratio n and suggest ways in which this situation can be detected and remedie d. As an example, we consider the method of lines (MOL) solution of Ri chards' equation (RE), which models ow through variably saturated poro us media. When the solution to this problem has a sharp moving front, and the Jacobian is even slightly ill conditioned, the corrector itera tion used in many integrators can terminate prematurely, leading to in correct results. While this problem can be solved by tightening the to lerances for the solvers used in the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver a nd/or recompute the Jacobian more frequently. Of these two, recomputat ion of the Jacobian is the more important. We propose a criterion base d on an estimate of the norm of the time derivative of the Jacobian fo r recomputation of the Jacobian and a second criterion based on a cond ition estimate for tightening of the termination criteria of the nonli near solver.