Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations

2005 ◽  
Vol 11 (7) ◽  
pp. 645-653 ◽  
Author(s):  
Ronald E. Mickens
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Fundamental Principle ◽  
Difference Schemes ◽  
Dynamic Consistency ◽  
Finite Difference Schemes ◽  
Nonstandard Finite Difference

AN INTRODUCTION TO NONSTANDARD FINITE DIFFERENCE SCHEMES

1999 ◽  
Vol 07 (01) ◽  
pp. 39-58 ◽  
Author(s):  
RONALD E. MICKENS
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Difference Schemes ◽  
Discrete Models ◽  
Finite Difference Schemes ◽  
Partial Differential ◽  
Numerical Instabilities ◽  
Nonstandard Finite Difference

Nonstandard finite difference schemes offer the potential for either constructing exact discrete models of differential equations or obtaining discrete models that do not have the elementary numerical instabilities. While the general laws for constructing such schemes are not precisely known at the present time, a number of important rules have been discovered. This paper provides an introduction to the nonstandard finite difference rules, explains their significance, and applies them to several model ordinary and partial differential equations. Several major unresolved issues and problems are briefly discussed.

scholarly journals Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1038 ◽  
Author(s):  
María Ángeles Castro ◽  
Miguel Antonio García ◽  
José Antonio Martín ◽  
Francisco Rodríguez
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Linear Delay ◽  
Delay Differential Systems ◽  
Delay Differential ◽  
Consistency Properties ◽  
Nonstandard Finite Difference

In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations X ′ ( t ) = A X ( t ) + B X ( t - τ ) , where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods.

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