In this paper, we present a new pressure-correction projection scheme for solving the time-dependent Navier-Stokes equations, which is based on the Crank-Nicolson extrapolation method in the time discretization. Error estimates for the velocity and the pressure of semidiscretized scheme are derived by interpreting the projection scheme as second-order time discretization of a perturbed system w...

This article provides a practical overview of numerical solutions to the heat equation using the finite difference method. The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem involving the one-dimensional heat equation. Complete, working Matlab codes for each scheme are presented. The result...

In this paper we develop two novel numerical methods for the partial integral differential equation arising from the valuation of an option whose underlying asset is governed by a jump diffusion process. These methods are based on a fitted finite volume method for the spatial discretization, an implicit-explicit time stepping scheme and the Crank-Nicolson time stepping method. We show that the ...

This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain Ω with artificial boundary conditions set on the arbitrarily shaped boundary of Ω. These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. Af...

. In this paper, we develop a quadratic spline collocation method for integrating the nonlinear partial differential equations (PDEs) of a plug flow reactor model. The method is proposed in order to be used for the operation of control design and/or numerical simulations. We first present the Crank-Nicolson method to temporally discretize the state variable. Then, we develop and analyze the pro...

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 give a highly efficient, accurate and unconditionally stable algorithm to solve the partial differential equation for simulating the action potential propagation through cardiac tissue. In the new algorithm, we discretize the space domain by combining a compact finite difference scheme with an alternating direction implicit (ADI) scheme, which has fourth-order accuracy for interior mesh poin...

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficient on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann-Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the...