The Crank–Nicolson Galerkin method and convergence for the time-dependent Maxwell-Dirac system under the Lorentz gauge

2021 ◽  
pp. 114007
Author(s):  
Yaoyao Fu ◽  
Liqun Cao
Keyword(s):  
Galerkin Method ◽  
Time Dependent ◽  
Dirac System ◽  
Lorentz Gauge ◽  
Crank Nicolson

scholarly journals A Galerkin Method for a Gaseous Ignition Model

2012 ◽  
Vol 17 (2) ◽  
pp. 224
Author(s):  
M. Salman ◽  
Jintae Kim
Keyword(s):  
Error Estimate ◽  
Galerkin Method ◽  
Integrodifferential Equation ◽  
Order Of Convergence ◽  
Combustion Model ◽  
Gas Combustion ◽  
Galerkin Procedure ◽  
Variational Form ◽  
The Given ◽  
Crank Nicolson

We consider a Galerkin procedure to solve a parabolic integrodifferential equation that arises in a gas combustion model. This model has been proposed by Kassoy and Poland, and subsequently analyzed by Bebernes, Eberly and Bressan. The problem is formulated in the variational form. In order to estimate the error, some intermediate projection has been employed. Under certain conditions on the given data, the error estimate has been obtained. A fully descretized version by using an extrapolated Crank-Nicolson method has been applied and the order of convergence  derived.  

scholarly journals Computational Sinc-scheme for extracting analytical solution for the model Kuramoto-Sivashinsky equation

2019 ◽  
Vol 9 (5) ◽  
pp. 3720
Author(s):  
Kamel Al-Khaled ◽  
Issam Abu-Irwaq
Keyword(s):  
Analytical Solution ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Present Article ◽  
Point Theory ◽  
Algebraic Equations ◽  
Time Direction ◽  
Numerical Examples ◽  
Sivashinsky Equation ◽  
Crank Nicolson

The present article is designed to supply two different numerical<br />solutions for solving  Kuramoto-Sivashinsky equation. We have made<br />an attempt to develop a numerical solution via the use of<br />Sinc-Galerkin method for  Kuramoto-Sivashinsky equation, Sinc<br />approximations to both derivatives and indefinite integrals reduce<br />the solution to an explicit system of algebraic equations. The fixed<br />point theory is used to prove the convergence of the proposed<br />methods. For comparison purposes, a combination of a Crank-Nicolson<br />formula in the time direction, with the Sinc-collocation in the<br />space direction is presented, where the derivatives in the space<br />variable are replaced by the necessary matrices to produce a system<br />of algebraic equations. In addition, we present numerical examples<br />and comparisons to support the validity of these proposed<br />methods.

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