Stability Theory for Partial Difference Operators

SIAM Review ◽  
1969 ◽  
Vol 11 (2) ◽  
pp. 152-195 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Stability Theory ◽  
Difference Operators ◽  
Partial Difference

On divided difference operators in function algebras

2008 ◽  
Vol 41 (2) ◽  
Author(s):  
Piotr Multarzyński
Keyword(s):  
Divided Difference ◽  
Difference Operators ◽  
Difference Quotient ◽  
Partial Difference ◽  
Function Algebras ◽  
Definition Of ◽  
Product Rules ◽  
Set Of Points ◽  
Tension Structure

AbstractIn this paper we study divided difference operators of any order acting in function algebras. In the definition of difference quotient operators we use a tension structure defined on the set of points on which depend the functions of the algebras considered. In the paper we mention the oportunity for partial difference quotient operators as well as for some purely algebraic definition of divided difference operators in terms of the suitable Leibniz product rules.

scholarly journals ON THE CONSTRUCTION OF PARTIAL DIFFERENCE SCHEMES II: DISCRETE VARIABLES AND SCHWARZIAN LATTICES

2016 ◽  
Vol 56 (3) ◽  
pp. 236 ◽  
Author(s):  
Decio Levi ◽  
Miguel A. Rodriguez
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Open Problem ◽  
Burgers Equation ◽  
Difference Schemes ◽  
Difference Operators ◽  
Discrete Variables ◽  
Partial Differential ◽  
Partial Difference ◽  
Arbitrary Lattice

In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing a partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut–Schwarz–Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut–Schwarz–Young theorem. A discussion on the numerical results is presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.

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