Abstract—A physical model for guiding the wave in photorefractive media is studied. Propagation of cos-Gaussian beam as the special cases of sinusoidal-Gaussian beams in photorefractive crystal is simulated numerically by the Crank-Nicolson method in one dimension. Results show that the beam profile deforms as the energy transfers from the center to the tails under propagation. This simulation ...

This paper provides an error analysis for the Crank–Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier–Stokes problem, where the finite element space pair (Xh,Mh) for the approximation (uh, p n h) of the velocity u and the pressure p is constructed by the low-order finite ele...

The study of convective heat and mass transfer flow over a vertical plate with nth order chemical reaction in a porous medium is analyzed both analytically and numerically. The resulting governing boundary layer equations are highly non-linear and coupled form of partial differential equations and have been solved by using implicit finite difference method of Crank-Nicolson. To check the accura...

In this work we study a preconditioned iterative method for some higher–order time discretizations of linear parabolic partial differential equations. We use the Padé approximations of the exponential function to discretize in time and show that standard solution algorithms for lower–order time discretization schemes, such as Crank–Nicolson and implicit Euler, can be reused as preconditioners f...

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank–Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full discretizations, we employ elliptic reconstructions...

This paper is concerned with transparent boundary conditions (TBCs) for wide angle “parabolic” equations (WAPEs) in underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs introduce slight numerical reflections at this artificial boundary and also render the overall Crank–Nicolson finite difference method only conditionally stable. Here, a novel discrete TBC...

Three new fully implicit methods which are based on the (5,5) Crank–Nicolson method, the (5,5) N-H (Noye–Hayman) implicit method and the (9,9) N-H implicit method are developed for solving the heat equation in two dimensional space with non-local boundary conditions. The latter is fourth-order while the others are second-order. While the implicit methods developed here, like the scheme based on...

Many temporal discretization methods for linear evolution equations converge uniformly on compact time intervals at the rate 1 nα only for sufficiently smooth initial data. It is shown that these methods can be regularized such that the new schemes converge ‘in the average’ at the rate 1 nα for all initial data. Examples given include the Crank-Nicholson scheme and the alternating direction imp...

The stability is one of the most basic requirement for the numerical model, which is mostly elaborated for the linear problems. In this paper we analyze the stability notions for the nonlinear problems. We show that, in case of consistency, both the N-stability and Kstability notions guarantee the convergence. Moreover, by using the Nstability we prove the convergence of the centralized Crank--...