Finite difference method

Abstract: SUMMARY A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function-vorticity form of the governing equations was approximated by upwind or central finite-differences. Both types of finite-difference approximations were solved by a multi-grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0 1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.

Abstract: SUMMARY We previously derived a general criterion for optimally accurate numerical operators for the calculation of synthetic seismograms in the frequency domain (Geller & Takeuchi 1995). We then derived modi¢ed operators for the Direct Solution Method (DSM) (Geller & Ohminato 1994) which satisfy this general criterion, thereby yielding signi¢cantly more accurate synthetics (for any given numerical grid spacing) without increasing the computational requirements (Cummins et al .1 994; Takeuchi, Geller & Cummins 1996; Cummins, Takeuchi & Geller 1997). In this paper, we derive optimally accurate time-domain ¢nite diierence (FD) operators which are second order in space and time using a similar approach. As our FD operators are local, our algorithm is well suited to massively parallel computers. Our approach can be extended to other methods (e.g. pseudo-spectral) for solving the elastic equation of motion. It might also be possible to extend this approach to equations other than the elastic equation of motion, including non-linear equations.

Abstract: We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional ellip- tic partial differential equations. A prototype is the convection-diffusion equa- tion. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, com- bined with a reordering and a block iterative technique for solving the reduced system. For constant-coefficient problems, we present analytic bounds on the spectral radii of the iteration matrices in terms of cell Reynolds numbers that show the methods to be rapidly convergent. In addition, we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the "unreduced" system.

Abstract: Two basic refinements of the finite-difference method for 3-D dc resistivity modeling are presented. The first is a more accurate formula for the source singularity removal. The second is the analytic computation of the source terms that arise from the decomposition of the potential into the primary potential because of the source current and the secondary potential caused by changes in the electrical conductivity. Three examples are presented: a simple two-layered model, a vertical contact, and a buried sphere. Both accurate and approximate Dirichlet boundary conditions are used to compute the secondary potential. Numerical results show that for all three models, the average percentage error of the apparent resistivity obtained by the modified finite-difference method with accurate boundary conditions is less than 0.5%. For the vertical contact and the buried sphere models, the error caused by the approximate boundary condition is less than 0.01%.