Stone method

Abstract: In this paper, a numerical Laplace transform algorithm which is based on the decomposition method is introduced for the approximate solution of a class of nonlinear differential equations. The technique is described and illustrated with some numerical examples. The results assert that this scheme is rapidly convergent and quite accurate by which it approximates the solution using only few terms of its iterative scheme.

Abstract: In this paper, we present a sequence of iterative methods improving Newton's method for solving nonlinear equations. The Adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses. The orders of convergence are derived analytically, and then rederived by applying symbolic computation of Maple. Some numerical illustrations are given.

Abstract: This repprt documents a computer code for solving problems of variably saturated, single-phase flow in porous media. The mathematical model of this physical process is developed by combining the law of conservation of fluid mass with a nonlinear form of Darcy's law. The resultant mathematical model, or flow equation, is written with total hydraulic potential as the dependent variable. This allows straightforward treatment of both saturated and unsaturated conditions. The spatial derivatives in the flow equation are approximated by central differences written about grid-block boundaries. Time derivatives are approximated by a fully implicit backward scheme. Nonlinear storage terms are linearized by an implicit Newton-Raphson method. Nonlinear conductance terms, boundary conditions, and sink terms are linearized implicitly. Relative hydraulic conductivity is evaluated at cell boundaries by using full upstream weighting, the arithmetic mean, or the geometric mean of values from adjacent cells. Saturated hydraulic conductivities are evaluated at cell boundaries by using distance-weighted harmonic means. The linearized matrix equations are solved using the strongly implicit procedure. Nonlinear conductance and storage coefficients are assumed to be represented by one of three closed-form algebraic equations. Alternatively, these values may be interpolated from tabulated data. Nonlinear boundary conditions treated by the code include infiltration, evaporation, and seepage faces. Extraction by plant roots is included as a nonlinear sink term. The code is written in standard ANSI Fortran. Extensive use of subroutines and function subprograms provides a modular code that is easily modified. A complete listing of data-input requirements and input and output for a one-dimensional infiltration problem and for a two-dimensional problem involving infiltration, evaporation, and evapotranspiration (plant-root extraction) are included.