An implicit energy-decaying modified Crank--Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem $H^2$ estimates discretized hyperbolic--parabolic system. For practical computation, further in space, resulting a fully discrete fini...

We consider a model initial and boundary value problem for the wide-angle ‘parabolic’ equation Lur = icu of underwater acoustics, where L is a second-order differential operator in the depth variable z with depthand range-dependent coefficients. We discretize the problem by the Crank–Nicolson finite difference scheme and also by the forward Euler method using nonuniform partitions both in depth...

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 develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, tim...

Transparent boundary conditions (TBCs) for general Schrödinger– type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are non–local in time (of memory type). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite d...

We propose and analyze a partitioned numerical method for the fully evolutionary Stokes-Darcy equations that model the coupling between surface and groundwater flows. The proposed method uncouples the surface from the groundwater flow by using the implicit-explicit combination of the Crank-Nicolson and Leapfrog methods for the discretization in time with added stabilization terms. We prove that...

The standard ‘parabolic’ approximation to the Helmholtz equation is used in order to model long-range propagation of sound in the sea in the presence of cylindrical symmetry in a domain with a rigid bottom of variable topography. The rigid bottom is modeled by a homogeneous Neumann condition and a paraxial approximation thereof proposed by Abrahamsson and Kreiss. The resulting initial-boundary-...

In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H . We consider two cases. If H > 1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in...