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This paper concerns the efficient numerical solution of the time-dependent, three-dimensional Heston-Hull-White PDE for the fair prices of European call options. The numerical solution method described in this paper consists of a finite difference discretization on non-uniform spatial grids followed by an Alternating Direction Implicit scheme for the time discretization and extends the method recently proved effective by In’t Hout & Foulon (2010) for the simpler, two-dimensional Heston PDE. [ABSTRACT FROM AUTHOR]

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]

The numerical valuation of American put options under the Heston stochastic volatility model is considered. We investigate in this paper the potential of combining the recent splitting approach of Ikonen & Toivanen (2004, 2009) with Alternating Direction Implicit schemes to obtain more efficient numerical methods. [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]

The stability of ADI type schemes is considered when applied to two- and three-dimensional pure diffusion equations with mixed derivative terms. We present a first stability result that takes the actual size of the mixed derivative coefficients into account. [ABSTRACT FROM AUTHOR]

By exploiting the fact that conservation laws form the kernel of a discrete Euler operator, we use a recently introduced symbolic-numeric approach to construct a new class of finite difference methods for the modified Korteweg-de Vries (mKdV) equation, that preserve the local conservation laws of mass and energy. [ABSTRACT FROM AUTHOR]

In this work, three finite difference methods have been used to solve a three dimensional advection-diffusion equation with given initial and boundary conditions. The three methods are fourth order finite difference method, Crank-Nicolson and Implicit Chapeau Function. We compare the performance of the methods by computing L2 error, L∞ error and some performance indices such as mass distribution ratio (MDR), mass conservation ratio (MCR), total mass and R² which is a measure of total variation in particle distribution. [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]