@article{tocci_kelley_miller_kees_1998, title={Inexact Newton methods and the method of lines for solving Richards' equation in two space dimensions}, volume={2}, ISSN={["1420-0597"]}, DOI={10.1023/A:1011562522244}, number={4}, journal={COMPUTATIONAL GEOSCIENCES}, author={Tocci, MD and Kelley, CT and Miller, CT and Kees, CE}, year={1998}, pages={291–309} }
 @article{miller_williams_kelley_tocci_1998, title={Robust solution of Richards' equation for nonuniform porous media}, volume={34}, ISSN={["0043-1397"]}, DOI={10.1029/98WR01673}, abstractNote={Capillary pressure–saturation‐relative permeability relations described using the van Genuchten [1980] and Mualem [1976] models for nonuniform porous media lead to numerical convergence difficulties when used with Richards' equation for certain auxiliary conditions. These difficulties arise because of discontinuities in the derivative of specific moisture capacity and relative permeability as a function of capillary pressure. Convergence difficulties are illustrated using standard numerical approaches to simulate such problems. We investigate constitutive relations, interblock permeability, nonlinear algebraic system approximation methods, and two time integration approaches. An integral permeability approach approximated by Hermite polynomials is recommended and shown to be robust and economical for a set of test problems, which correspond to sand, loam, and clay loam media.}, number={10}, journal={WATER RESOURCES RESEARCH}, author={Miller, CT and Williams, GA and Kelley, CT and Tocci, MD}, year={1998}, month={Oct}, pages={2599–2610} }
 @article{kelley_miller_tocci_1998, title={Termination of Newton/Chord iterations and the method of lines}, volume={19}, ISSN={["1064-8275"]}, DOI={10.1137/S1064827596303582}, abstractNote={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 In this paper, we show how an out-of-date Jacobian coupled with moderate ill conditioning can lead to premature termination of the corrector iteration and suggest ways in which this situation can be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.}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Kelley, CT and Miller, CT and Tocci, MD}, year={1998}, month={Jan}, pages={280–290} }
 @article{tocci_kelley_miller_1997, title={Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines}, volume={20}, ISSN={["0309-1708"]}, DOI={10.1016/S0309-1708(96)00008-5}, abstractNote={The pressure-head form of Richards' equation (RE) is difficult to solve accurately using standard time integration methods. For example, mass balance errors grow as the integration progresses unless very small time steps are taken. Further, RE may be solved for many problems more economically and robustly with variable-size time steps rather than with a constant time-step size, but variable step-size methods applied to date have relied upon empirical approaches to control step size, which do not explicitly control temporal truncation error of the solution. We show how a differential algebrain equation implementation of the method of lines can give solutions to RE that are accurate, have good mass balance properties, explicitly control temporal truncation error, and are more economical than standard approaches for a wide range of solution accuracy. We detail changes to a standard integrator, DASPK, that improves efficiency for the test problems considered, and we advocate the use of this approach for both RE and other problems involving subsurface flow and transport phenomena.}, number={1}, journal={ADVANCES IN WATER RESOURCES}, author={Tocci, MD and Kelley, CT and Miller, CT}, year={1997}, month={Feb}, pages={1–14} }