where f(t),g(t), and h(t) are known continuous functions of t in the interval (0,1). Here N(u) is a nonlinear function of u. Let the above equation be singular at these two boundary value points t= 0,1. Scientists and engineers are interested in singular BVPs because they arise in a wide range of applications, such as in chemical engineering, mechanical engineering, nuclear industry, and nonlin...

In the present work we derive and study a non-linear elliptic PDE coming from the problem of estimation of sound speed inside the Earth. The physical setting of the PDE allows us to pose only a Cauchy problem, and hence is ill-posed. However, we are still able to solve it numerically on a long enough time interval to be of practical use. We used two approaches. The first approach is a finite di...

Many applications based on finite element and finite difference methods include the solution of large sparse linear systems using preconditioned iterative methods. Matrix vector multiplication is one of the key operations that has a significant impact on the performance of any iterative solver. In this paper, recent developments in sparse storage formats on vector machines are reviewed. Then, s...

An interpolation polynomial of order N is constructed from p indepen­ dent subpolynomials of order n '" Nip. Each such subpolynomial is found independently and in parallel. Moreover, evaluation of the polynomial at any given point is done independently and in parallel, except for a final step of summation of p elements. Hence, the algorithm has almost no commu­ ,:.. nication overhead and can be...

We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrödinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove second-order error es...

by, for example, a finite difference method in the computational (μ, χ, φ) space. The above standard dipole coordinates is convenient and certainly appropriate for analytical studies in which the Earth’s dipolar field plays central roles. It also works as a base coordinates for the node and cell generation of the finite element method in the dipole geometry [2, 3]. However, when one tries to us...

In this paper we present a stable hybrid scheme for viscous problems. The hybrid method combines the unstructured finite volume method with high-order finite difference methods in complex geometries. The coupling procedure between the two numerical methods is based on energy estimates and stable interface conditions are constructed. Numerical calculations show that the hybrid method is efficien...

We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and L norm error bounds. These bound...

A finite element method for the 1-periodic Korteweg-de Vries equation "t + 2uux + "xxx = ° is analyzed. We consider first a semidiscrete method (i.e., discretization only in the space variable), and then we analyze some unconditionally stable fully discrete methods. In a special case, the fully discrete methods reduce to twelve point finite difference schemes (three time levels) which have seco...

Abstract: This paper presents some numerical methods for Allen-Cahn equation using different time stepping and space discretization methods with non-periodic boundary conditions. In space the equation is discretized by Chebyshev spectral method, while in time the exponential time differencing fourth-order Runge-Kutta (ETDRK4) and implicit-explicit scheme is used. For comparison we also use the ...