6 research outputs found
The numerical solution of two-dimensional moving boundary problems using curvilinear co-ordinate transformations
A numerical method is described for the solution of two-dimensional moving boundary problems by tansforming the curved, fixed and moving boundaries in the originalco-ordinate system (x,y) into an orthogonal or, in general, nonorthogonal curvilinear system (ξ,η) such that the curved boundaries become (ξ,η) co-ordinate lines. All computations are then carried out in the transformed region using a fixed, rectangular (ξ,η) mesh which corresponds to a moving, non-rectangular (x,y) mesh. A one-phase, two-dimensional problem is solved by using two different such transformations and the results are compared with those from finite-element, enthalpy and isotherm migration methods
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The numerical solution of elliptic and parabolic partial differential equations with boundary singularities
A general numerical method is described for the solution of linear elliptic and parabolic partial differential equations in the presence of boundary singularities. The method is suitable for use with either a finite-difference or finite element scheme. Modified approximations for the derivatives are developed using the local analytical form of the singularity. General guidelines are given showing how the local analytical form can be found and how the modified approximations can be developed for many problems of mathematical physics. These guidelines are based on the reduction of the differential equation to the form Δu = gu + f. The potential problem treated by Motz and Woods is taken as a numerical example. The numerical results compare favourably with those obtained by other techniques
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The treatment of boundary singularities in axially symmetric problems containing discs
Axially symmetric problems (e.g. Laplace's equation in cylindrical co-ordinates) containing discs possess boundary singularities arising from the mixed boundary conditions that occur across the disc edge. A modified finite-difference method is presented which effectively eliminates the inaccuracies that occur in the standard numerical solution near such singularities. Techniques for developing the analytical forms of such singularities are given and modified finite-difference approximations are obtained. The steady-state diffusion of oxygen around a circular electrode is taken as the model problem and a modified quadrature method is presented for the calculation of the oxygen flux through the electrode
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A survey of the formulation and solution of free and moving boundary (stefan) problems
Modified finite-difference approximations near the singularitiy in the problem of motz
A simple, modified finite-difference method is described for solving Laplace's equation with boundary singularities of the infinite derivative type. Modified approximations for the derivatives of the Laplacian equation are employed near the singularity. These are developed from a truncated series form of the local analytical solution. The method is applied to the problem of Motz. The numerical results compare favourably with those obtained by other techniques