An integrated finite difference algorithm is presented for numerically solving the governing equation of saturated‐unsaturated flow in deformable porous media. In recognition that stability of the explicit equation is a local phenomenon a mixed explicit‐implicit procedure is used for marching in the time domain. In this scheme the explicit changes in potential are first computed for all elements in the system, after which implicit corrections are made only for those elements for which the stable time step is less than the time step being used. Time step sizes are automatically controlled in order to optimize the number of iterations, to control maximum change in potential during a time step, and to obtain desired outputs. Time derivatives, estimated on the basis of system behavior during two previous time steps, are used to start the iteration process and to evaluate nonlinear coefficients. Boundary conditions and sources can vary with time or with the dependent variable. Input data are organized into convenient blocks. Accuracy of solutions can be affected by modeling errors, different types of truncation errors, and convergence errors. The algorithm constitutes an efficient tool for analyzing linear and nonlinear fluid flow problems in multidimensional heterogeneous porous media with complex geometry. An important limitation is that the model cannot conveniently handle arbitrary anisotropy and other general tensorial quantities.

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Ahstract-We explore the relationship of Monaghan's version of "smoothed-particle hydro dynamics," here called "smoothed-particle applied mechanics," to nonequilibrium molecular dynam ics. We first use smoothed particles to model the simplest possible linear transport problems, as well as a liquid-drop problem. We then consider both gas-phase and dense-fluid versions of Rayleigh-Benard convection, all in two space dimensions. We also discuss the possibility of combining the microscopic and macroscopic techniques in a hybrid scheme well-suited to the massively-parallel modelling of large-scale nonequilibrium flows.

SUMMARYIn Part I' of this paper we have established local stability and convergence criteria for the mixed explicitimplicit finite element scheme and have shown that the proposed iterative method converges under certain conditions. Part I1 describes various practical aspects of the solution strategy such as convergence criteria for terminating the iterations, automatic control of time step size, reclassification of nodes from explicit to implicit during execution, estimation of time derivatives, and automatic adjustment of the implicit weight factor. Several examples are included to demonstrate certain aspects of the theory and illustrate the capabilities of the new approach.

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