Numerical Method of Integrating the Variational Equations for Cauchy Problem Based on Differential Transformations

2015 ◽  
Vol 47 (9) ◽  
pp. 63-75 ◽  
Author(s):  
Mikhail Yu. Rakushev
Author(s):  
Blaise Faugeras ◽  
Amel Ben Abda ◽  
Jacques Blum ◽  
Cedric Boulbe

International audience A numerical method for the computation of the magnetic flux in the vacuum surrounding the plasma in a Tokamak is investigated. It is based on the formulation of a Cauchy problem which is solved through the minimization of an energy error functional. Several numerical experiments are conducted which show the efficiency of the method.


2018 ◽  
Vol 52 (3) ◽  
pp. 803-826 ◽  
Author(s):  
H.A. Erbay ◽  
S. Erbay ◽  
A. Erkip

In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. The most important characteristic of the numerical method is that it is directly applied to the nonlocal equation by introducing the discrete convolution operator. Starting from the continuous Cauchy problem defined on the real line, we first construct the discrete Cauchy problem on a uniform grid of the real line. Thus the semi-discretization in space of the continuous problem gives rise to an infinite system of ordinary differential equations in time. We show that the initial-value problem for this system is well-posed. We prove that solutions of the discrete problem converge uniformly to those of the continuous one as the mesh size goes to zero and that they are second-order convergent in space. We then consider a truncation of the infinite domain to a finite one. We prove that the solution of the truncated problem approximates the solution of the continuous problem when the truncated domain is sufficiently large. Finally, we present some numerical experiments that confirm numerically both the expected convergence rate of the semi-discrete scheme and the ability of the method to capture finite-time blow-up of solutions for various convolution kernels.


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