In this study, the finite difference method is applied to an optimal control problem controlled by two functions which are in the coefficients of two-dimensional Schrodinger equation. Convergence of the finite difference approximation according to the functional is proved. We have used the implicit method for solving the two-dimensional Schrodinger equation. Although the implicit scheme obtaine...

It has been previously shown that the temporal integration of hyperbolic partial differential equations may , because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic p.d.e's (linear and non-linear) whose boundary treatment is done via t...

An easy-to-use variant of the finite difference method (FDM), the polynomial finite difference method (PFDM), for the numerical solution of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs), is presented. Compared to the traditional implementation of the FDM, the PFDM approach has two major advantages: straightforward implementation, and easily adjustable accura...

Using classical finite difference schemes often generates numerical drawbacks such as spurious oscillations in the solution of the famous Black–Scholes partial differential equation. We analyze the fully implicit scheme, frequently used numerical method in Finance, that in the presence of discontinuous payoff and low volatility arises spurious oscillations. We propose a modification of this sch...

Based on the concept of domain decomposition we construct a class of alternating group explicit method for fourth order parabolic equations. Furthermore, an exponential type alternating group explicit method for 2D convection-diffusion equations is derived, which is effective in convection dominant cases. Both of the two methods have the property of unconditional stability and intrinsic paralle...

A relaxation system based on a Lattice-Boltzmann type discrete velocity model is considered in the low Mach number limit. A third order relaxation scheme is developed working uniformly for all ranges of the mean free path and Mach number. In the incompressible Navier-Stokes limit the scheme reduces to an explicit high order finite difference scheme for the incompressible Navier-Stokes equations...

Numerical simulation of advective-dispersive contaminant transport is carried out by using high-order compact finite difference schemes combined with second-order MacCormack and fourth-order Runge-Kutta schemes. Both of the two schemes have accuracy of sixth-order in space. A sixth-order MacCormack scheme is proposed for the first time within this study. For the aim of demonstrating efficiency ...

Spurious long-term solutions of a finite-difference method for a hyperbolic conservation law with a general nonlinear source term are studied. Results are contrasted with those that have been established for nonlinear ordinary differential equations. Various types of spurious behavior are examined, including spatially uniform equilibria that exist for arbitrarily small time-steps, nonsmooth ste...

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter 0 < ε ≪ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagat...

An explicit/implicit domain decomposition method is applied to the time-dependent heatconduction problem in a 2-d, strongly anisotropic medium (a magnetized plasma), using a formulation of the spatial derivatives which avoids the pollution of perpendicular by parallel heat fluxes. The time-stepping at the sub-domain boundaries is done using a DuFort-Frankel scheme, which leads to a time step li...