Numerical Methods for First-Order ODEs

2020 ◽  
pp. 41-112
Author(s):  
Sulaymon L. Eshkabilov
Keyword(s):  
Numerical Methods ◽  
First Order

scholarly journals Note on the Sensitivity of Simulated Solutions of the Nonlinear Singularly Perturbed Dynamical System on the Used Different Rewriting into a System of First Order Differential Equations

2016 ◽  
Vol 24 (39) ◽  
pp. 51-55
Author(s):  
Vladimír Liška ◽  
Zuzana Šútova ◽  
Dušan Pavliak
Keyword(s):  
Dynamical System ◽  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Numerical Scheme ◽  
Singularly Perturbed ◽  
First Order ◽  
First Order Differential Equations

Abstract In this paper we analyze the sensitivity of solutions to a nonlinear singularly perturbed dynamical system based on different rewriting into a System of the First Order Differential Equations to a numerical scheme. Numerical simulations of the solutions use numerical methods implemented in MATLAB.

HOMOGENIZATION OF FIRST-ORDER EQUATIONS WITH u/∊-PERIODIC HAMILTONIAN: RATE OF CONVERGENCE AS ∊ → 0 AND NUMERICAL METHODS

2011 ◽  
Vol 21 (06) ◽  
pp. 1317-1353 ◽  
Author(s):  
YVES ACHDOU ◽  
STEFANIA PATRIZI
Keyword(s):  
Numerical Methods ◽  
Rate Of Convergence ◽  
Effective Hamiltonian ◽  
First Order ◽  
Jacobi Equations ◽  
Homogenization Problems

We consider homogenization problems for first-order Hamilton–Jacobi equations with u∊/∊ periodic dependence, recently introduced by Imbert and Monneau, and also studied by Barles: this unusual dependence leads to nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter ∊ tends to 0. We obtain the same rates as those obtained by Capuzzo Dolcetta and Ishii for the more usual homogenization problems without the dependence in u∊/∊. In the second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

scholarly journals Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation

2019 ◽  
Vol 53 (2) ◽  
pp. 443-473 ◽  
Author(s):  
Philippe Chartier ◽  
Loïc Le Treust ◽  
Florian Méhats
Keyword(s):  
Numerical Methods ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Splitting Methods ◽  
Convergence Behavior ◽  
First Order ◽  
Time Splitting ◽  
Uniform Accuracy

This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schrödinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.

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