The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed model ordinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. F...

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed model ordinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. F...

A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the r...

In this article, we present a novel technique for boosting the order of accuracy of formally low-order finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution. This technique allows us to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Thro...

In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind fini...

Many numerical methods have been developed for the solution of Poisson and Laplace equations such as finite-difference methods, finite-element methods, boundary-element methods, etc. Sinc numerical methods developed recently by Stenger and co-workers excel over other methods for problems involving singularities, infinite or semi-infinite domains as well as boundary layer behaviors. In this pape...

High-frequency solutions of one or several Schrödinger-type equations are well known to differ very little from the plane wave solutions exp[±ikx]. That is, the potential terms impact the envelope of a high-frequency plane wave by only a small amount. However, when such equations are solved by a finite-difference method, the highest-frequency solutions may, under certain conditions, turn out to...

A wide family of finite-difference methods for the linear advection equation, based on a six-point stencil, is presented. The family depends on three parameters and includes most of the classical linear schemes. A stability and consistency analysis is carried out. Numerical examples show the performance of the different methods according to the choice of the parameters. The problem of the deter...

We present a fourth order finite difference method for doubly singular boundary value problem p x y′ x ′ q x f x, y , 0 < x ≤ 1 with boundary conditions y 0 A or y′ 0 0 or limx→ 0p x y′ x 0 and αy 1 βy′ 1 γ , where α > 0 , β ≥ 0 , γ and A are finite constants. Here p 0 0 and q x is allowed to be discontinuous at the singular point x 0. The method is based on uniform mesh. The accuracy of the me...

A new 9-point sixth-order accurate compact finite-difference method for solving the Helmholtz equation in one and two dimensions, is developed and analyzed. This scheme is based on sixth-order approximation to the derivative calculated from the Helmholtz equation. A sixth-order accurate symmetrical representation for the Neumann boundary condition was also developed. The efficiency and accuracy...