Finite-Difference Schemes for Partial Differential Equations

2006 ◽  
pp. 307-426
Author(s):  
Victor S. Ryaben’kii ◽  
Semyon V. Tsynkov
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Partial Differential

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.

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