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...

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 ...

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...

In this work, a space-time multigrid method which uses standard coarsening in both temporal and spatial domains and combines the use of different smoothers is proposed for the solution of the heat equation in one and two space dimensions. In particular, an adaptive smoothing strategy, based on the degree of anisotropy of the discrete operator on each grid-level, is the basis of the proposed mul...

Abstract. We study the error estimates for the alternating evolution discontinuous Galerkin (AEDG) method to one dimensional linear convection-diÆusion equations. The AEDG method for general convection-diÆusion equations was introduced in [H. Liu, M. Pollack, J. Comp. Phys. 307: 574–592, 2016], where stability of the semi-discrete scheme was rigorously proved for linear problems under a CFL-lik...