A unified framework to derive optimized compact schemes for a uniform grid is presented. The scheme coefficients are determined analytically by solving an optimization problem minimize the spectral error subject equality constraints that ensure specified order of accuracy. rigorous stability analysis also We show other types e.g., spatially explicit and biased finite differences, can be generat...

We study compact finite difference methods for the Schrödinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function ψ and external potential V(x). The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h4+τ2) in discrete l2 ,H1 and l norms with ...

We introduce the standard fourth order compact finite difference formulae. We show how these formulae apply in the special case of the heat equation. It is well known that the American option pricing problem may be formulated in terms of the Black Scholes partial differential equation (PDE) together with a free boundary condition. Standard methods allow this problem to be transformed into a mov...

Several numerical schemes utilizing central difference approximations have been developed to solve the Goursat problem. However, in a recent years compact discretization methods which leads to high-order finite difference schemes have been used since it is capable of achieving better accuracy as well as preserving certain features of the equation e.g. linearity. The basic idea of the new scheme...

Ultrasonic measuring technique is widely using for diagnosis of osteoporosis because of its safety. A precise simulation, however, is not reported, because the behavior of ultrasonic pulse propagated in a human bone is very complicated. In this paper, ultrasonic pulse propagation in bone phantom is analyzed using elastic Finite Difference Time Domain (FDTD) method. This proposed method treats n...

The multiscale multigrid method is presented in this article to solve the linear systems arising from a fourth order discretisation. We used a symbolic algebra packageMathematica to derive a family of finite difference approximations on a 27 point compact stencil. The unknown solution and its second derivatives are carried as unknowns at selected grid points. A set of test problems are presente...