Abstract. In recent years the usefulness of fast Laplace solvers has been extended to problems on arbitrary regions in the plane by the development of capacitance matrix methods.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Mathematics of Computation. Abstract. Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R . A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least O(h 55) in L2. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.1. Introduction. It is the purpose of this paper to develop some highly accurate finite difference methods for the Dirichlet problem for a general bounded region E2 in Rn. The most accurate of these has an L error of order at most h5 5, see Section 4. Our basic schemes use the standard (2n + 1)-point formula for the interior mesh points and are therefore only second order accurate. The increased accuracy is achieved by two steps of a deferred correction or Richardson extrapolation procedure. We also discuss the computer implementation of these methods in some detail.The use of deferred correction and Richardson extrapolation methods is justified by finding asymptotic expansions of the error. Wasow [20] has shown that no useful expansions of this kind exist if the boundary condition is approximated to a low order of accuracy. An obvious remedy for this problem, already mentioned by Wasow, is to use higher order interpolation or extrapolation formulas at any irregular mesh point, i.e. a mesh point in the open set E? which fails to have all its next neighbors in the closure of Q2. Volkov [19] proposed the use of high order one-dimensional Lagrange polynomials for this purpose. Because of the change of sign of the interpolation coefficients the matrix representing the difference scheme will then, in general, not be of positive type. The standard convergence proof based on a discrete maximum principle, (see Forsythe and Wasow [71) will therefore generally not apply. But by allowing the use of values of the mesh functions many mesh lengths away from the boundary, Volkov succeeded in designing schemes with diagonally dominant matrices. His
The numerical solution of a nonlinear hyperbolic equation not fulfilling the strict nonlinearity condition is considered. A solution, \ procedure is developed based on the random choice method, which permits the sharp tracking of discontinuities. As an illustration, an application to the two-phase flow of petroleum in underground reservoirs is presented .
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