Advection equations are an essential part of many mathematical models arising in different fields of science and engineering. It is important to treat such equations with efficient numerical schemes. The well-known Crank-Nicolson scheme will be applied. It will be shown that the accuracy of the calculated results can be improved when the Crank-Nicolson scheme is combined with the Richardson Ext...

The Crank–Nicolson method can be used to solve the Black–Scholes partial differential equation in one-dimension when both accuracy and stability is of concern. In multi-dimensions, however, discretizing the computational grid with a Crank–Nicolson scheme requires significantly large storage compared to the widely adopted Operator Splitting Method (OSM). We found that symmetrizing the system of ...

Implicit schemes have been extensively used in building physics to compute the solution of moisture diffusion problems in porous materials for improving stability conditions. Nevertheless, these schemes require important sub-iterations when treating nonlinear problems. To overcome this disadvantage, this paper explores the use of improved explicit schemes, such as Dufort–Frankel, Crank–Nicolson...

in this paper , the quintic b-spline collocation scheme is employed to approximate numerical solution of the kdv-like rosenau equation . this scheme is based on the crank-nicolson formulation for time integration and quintic b-spline functions for space integration . the unconditional stability of the present method is proved using von- neumann approach . since we do not know the exact solution...

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations wi...

The Crank–Nicolson scheme as well as its modified schemes is widely used in numerical simulations for the nonlinear Schrödinger equation. In this paper, we prove the multisymplecticity and symplecticity of this scheme. Firstly, we reconstruct the scheme by the concatenating method and present the corresponding discrete multisymplectic conservation law. Based on the discrete variational principl...

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.

This study proposes one-dimensional advection–diffusion equation (ADE) with finite differences method (FDM) using implicit spreadsheet simulation (ADEISS). By changing only the values of temporal and spatial weighted parameters with ADEISS implementation, solutions are implicitly obtained for the BTCS, Upwind and Crank–Nicolson schemes. The ADEISS uses iterative spreadsheet solution technique. ...

The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE mod...

We consider a linear, Schrödinger type p.d.e., the ‘Parabolic’ Equation of underwater acoustics, in a layer of water bounded below by a rigid bottom of variable topography. Using a change of depth variable technique we transform the problem into one with horizontal bottom, for which we establish an a priori H estimate and prove an optimal-order error bound in the maximum norm for a Crank-Nicols...