Abstract. This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in f...

Systems of nonlinear partial differential equations modeling turbulent fluid flow and similar processes present special challanges in numerical analysis. Regions of stability of implicit-explicit methods are reviewed, and an energy norm based on Dahlquist’s concept of G-stability is developed. Using this norm, a time-stepping Crank-Nicolson Adams-Bashforth 2 implicit-explicit method for solving...

We present a high resolution time integration method for the spherical harmonics equations based on the L-Trap method of Duraisamy, et al. This method is equivalent to the Crank-Nicolson method in smooth regions of the solution and changes form in non-smooth regions of the solution to avoid artificial oscillations in the solution. The method is nonlinear in the sense that the time integration u...

An efficient numerical self-consistent field theory (SCFT) algorithm is developed for treating structured polymers on spherical surfaces. The method solves the diffusion equations of SCFT with a pseudospectral approach that combines a spherical-harmonics expansion for the angular coordinates with a modified real-space Crank–Nicolson method for the radial direction. The self-consistent field equ...

Question 1 Détail des schémas. Schéma explicite : u k − uk ∆t = uk−1 − 2uk + uk+1 ∆x , en d’autres termes, en notant λ := ∆t/∆x, (1) u k = λu n k−1 + (1− 2λ)uk + λuk+1. (2) On sait d’après le cours que ce schéma est d’ordre 1 en temps et 2 en espace. Il est stable dans L sous condition CFL λ ≤ 1 2 . Schéma de Crank-Nicolson : u k − uk ∆t = 1 2 ( u k−1 − 2u n+1 k + u n+1 k+1 ∆x + uk−1 − 2uk + uk...

In this paper, we consider the time dependent Maxwell’s equations when dispersive media are involved. The Crank-Nicolson mixed finite element methods are developed for three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Optimal error estimates are proved for all three models solved by the Raviart-Thomas-Nédélec space...

In this paper, we develop the Crank-Nicolson finite difference method (C-N-FDM) to solve the linear time-fractional diffusion equation, formulated with Caputo’s fractional derivative. Special attention is given to study the stability of the proposed method which is introduced by means of a recently proposed procedure akin to the standard Von-Neumann stable analysis. Some numerical examples are ...

There have been several recent works on developing moving mesh methods for solving phase-field equations. However, it is observed that some of these moving mesh solutions are essentially different with the solutions on very fine fixed meshes. One of the purposes of this work is to understand the reason for the differences. We carried out numerical sensitivity studies systematically in this pape...

The effects of exponentially varying viscosity and linearly varying thermal conductivity on unsteady MHD natural convective flow past a semi infinite vertical plate in a thermally stratified medium is studied. The variables of both viscosity and thermal conductivity are considered only a function of temperature. The governing boundary layer equations of continuity, momentum and energy have been...

We develop an algorithm to solve tridiagonal systems of linear equations, which appear in implicit finite-difference schemes partial differential equations (PDEs), being the time-dependent Schr\"{o}dinger equation (TDSE) ideal candidate benefit from it. Our N-shaped partition method optimizes implementation numerical calculation on parallel architectures, without memory size constraints. Specif...