This paper is concerned with transparent boundary conditions for the one dimensional time–dependent Schrödinger equation. They are used to restrict the original PDE problem that is posed on an unbounded domain onto a finite interval in order to make this problem feasible for numerical simulations. The main focus of this article is on the appropriate discretization of such transparent boundary c...

The cylinder working fluid mean temperature, rate of heat fluxes to combustion chamber and temperature distribution on combustion chamber surface will be calculated in this research. By simulating thermodynamic cycle of engine, temperature distribution of combustion chamber will be calculated by the Crank-Nicolson method. An implicit finite difference method was used in this code. Special treat...

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...

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--...

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...

We analyze a Crank–Nicolson–type finite difference scheme for the Kuramoto– Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second–order error estimates.

The study of this paper is two-fold: On the one hand, we generalize the high-order local absorbing boundary conditions (LABCs) proposed in [J. Zhang, Z. Sun, X. Wu and D. Wang, Commun. Comput. Phys., 10 (2011), pp. 742–766] to compute the Schrödinger equation in the semiclassical regime on unbounded domain. We analyze the stability of the equation with LABCs and the convergence of the Crank-Nic...

In this paper, we develop a priori and a posteriori error estimates for wavelet-Taylor– Galerkin schemes introduced in Refs. 6 and 7 (particularly wavelet Taylor–Galerkin scheme based on Crank–Nicolson time stepping). We proceed in two steps. In the first step, we construct the priori estimates for the fully discrete problem. In the second step, we construct error indicators for posteriori esti...