Abstract. The problem of non-local nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with a nonlinearly coupled boundary value problem for a viscoelastic ‘pseudostress’ is considered (see, for example, DA Edwards in Z. angew. Math. Phys., 52, 2001, pp. 254—288). We present two numerical schemes using the implicit Euler method and also the Crank-Nicolson method. Each sc...

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain Í2 in Ä , with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting system of boundary value problems at each time step, a least-...

We consider the two-dimensional, time-dependent Schrödinger equation discretized with the Crank-Nicolson finite difference scheme. For this difference equation we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. We apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron in...

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time stepping procedure such as Crank-Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time stepping procedure. We show that minor modi cations in the AD and FS procedures can virtua...

This paper proposes the use of the Spectral method to simulate diffusive moisture transfer through porous materials, which can be strongly nonlinear and can significantly affect sensible and latent heat transfer. An alternative way for computing solutions by considering a separated representation is presented, which can be applied to both linear and nonlinear diffusive problems, considering hig...

In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analy...

the burgers’ equation is a simplified form of the navier-stokes equations that very well represents their non-linear features. in this paper, numerical methods of the adomian decomposition and the modified crank – nicholson, used for solving the one-dimensional burgers’ equation, have been compared. these numerical methods have also been compared with the analytical method. in contrast to...

in this article we discussed the numerical solution of Burgers’ equation using multigrid method. We used implicit method for time discritization and Crank-Nicolson scheme for space discritization for fully discrete scheme. For improvement we used Multigrid method in fully discrete solution. And also Multigrid method accelerates convergence of a basics iterative method by global correction. Nume...

We analyze the discretization of an initial-boundary value problem for the cubic Schrödinger equation in one space dimension by a Crank–Nicolson–type finite difference scheme. We then linearize the corresponding equations at each time level by Newton’s method and discuss an iterative modification of the linearized scheme which requires solving linear systems with the same tridiagonal matrix. We...