In this paper, a high-order method for solving the Schrodinger equation is introduced. We apply a compact finite difference approximation for discretizing spatial derivatives and we use the C-cubic spline collocation method for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. W...

The paper demonstrates advantages of using implicit finite differences for fast, accurate, and reliable differentiating and filtering of multidimensional signals defined on regular grids. In particular, applications to image enhancement and edge detection problems are considered. The theoretical contribution of the paper is threefold. The first adapts the Fourier-Padé-Galerkin approximations ap...

Keywords: Optimized scheme Explicit finite-difference Numerical dispersion Maximum norm Simulated annealing algorithm a b s t r a c t Conventional explicit finite-difference methods have difficulties in handling high-frequency components due to strong numerical dispersions. One can reduce the numerical dispersions by optimizing the constant coefficients of the finite-difference operator. Differ...

Numerical methods are important in computational neuroscience where complex nonlinear systems are studied. This report evaluates two methodologies, finite differences and Fourier series, for numerically integrating a nonlinear neural model based on a partial integro-differential equation. The stability and convergence criteria of four finite difference methods is investigated and their efficien...

Based on eight saul’yev asymmetry schemes and the concept of domain decomposition, a class of finite difference method (AGE) with intrinsic parallelism for 1D diffusion equations is constructed. Stability analysis for the method is done. We also pay attention to the implementation of the parallel algorithms for 2D convectiondiffusion equations. Based on another group of saul’yev asymmetry schem...

A high-order accurate, finite-difference method for the numerical solution of the incompressible Navier–Stokes equations is presented. Fourth-order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite-difference formulas. High-order accuracy in time is obtained by marching the solution with Runge–Kutta methods. The incom...

This paper is devoted with numerical solution of the system fractional differential equations (FDEs) which are generated by optimization problem using the Chebyshev collocation method. The fractional derivatives are presented in terms of Caputo sense. The application of the proposed method to the generated system of FDEs leads to algebraic system which can be solved by the Newton iteration meth...

1 In the present study, we applied a novel mesh-free method to solve acoustic wave equation. 2 Although the conventional finite difference methods determine the coefficients of its operator 3 based on the regular grid alignment, the mesh-free method is not restricted to regular 4 arrangements of calculation points. We derive the mesh-free approach using the 5 multivariable Taylor expansion. The...

Here f ∈ L(Ω) , g ∈ H(∂Ω) and Ω is a bounded domain in R with the smooth boundary ∂Ω ( see Figure 1 ). The method of lines for solving Problem I works well if Ω is a rectangular domain since the finite difference solution is expressed explicitly by use of eigenvalues and eigenvectors for the finite difference scheme([BGN70], [Nak65]). But one says that this method seems difficult to be applied ...

We prove a regularity property of finite difference schemes for the heat or diffusion equation ut = ∆u in maximum norm with large time steps. For a class of time discretizations including L-stable single-step methods and the second-order backward difference formula, with the usual second-order Laplacian, we show that solutions of the scheme gain first spatial differences boundedly, and also sec...