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This paper poses a magnetic field problem in cylindrical coordinate for two regions with different permeability for each one. The linear partial differential equation governing this problem is in the form of Laplace and Poisson equation. This problem is then solved using classical Gauss-Seidel Algorithm for the Finite difference (FD) solution of linear partial differential equation. In order to obtain adequate solution for a reasonable number of grid points for the regions under consideration a considerable amount of time will take for the program to converge. The paper presents a different technique known as Weighted Multi-Grid which will speed up the convergence process. In this method, the solution to the differential equation between two grid points for obtaining the initial condition is considered to be linear in nature with different weight for the value of each grid point. This problem is then solved for the minimum number of grid points let’s say one point in each region plus the boundary points. The initial guess for each new point for every level of computation is found as the weighted average of the two adjacent points. It then continues with finding the optimum weighting values for Laplace’s or Poisson’s equation. The main contribution is made by regarding the effect of the optimum initial weighted values of the variable vector in the convergence time for the Gauss-Seidel algorithm. [ABSTRACT FROM AUTHOR]

We construct a discrete fundamental solution for the parabolic Dirac operator which factorizes the non-stationary Schrödinger operator. With such fundamental solution we construct a discrete counterpart for the Teodorescu and Cauchy-Bitsadze operators and the Bergman projectors. We finalize this paper with convergence results regarding the operators and a concrete numerical example. [ABSTRACT FROM AUTHOR]

Two and three dimensional numerical investigations of viscous incompressible free surface flows are presented. The Reynolds averaged Navier-Stokes (RANS hereafter) equations are coupled with the continuity equation and solved by using a finite-difference method (FDM hereafter). A boundary fitted coordinate system is employed thereupon deficiencies associated mainly with interpolation at the free-surface that is still in wide are identified and addressed. Consequently, new expressions for boundary conditions at open frontiers are proposed. Third-order upwind schemes in which the third derivative of the wave elevation is intentionally introduced are used to determine the location of the particles laying on the free surface. Numerical wave reflections at boundaries are avoided by using artificial wave dampers there. [ABSTRACT FROM AUTHOR]