scholarly journals High-Order Compact Implicit Difference Methods For Parabolic Equations in Geodynamo Simulation

2009 ◽  
Vol 2009 ◽  
pp. 1-23 ◽  
Author(s):  
Don Liu ◽  
Weijia Kuang ◽  
Andrew Tangborn
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
High Order ◽  
Nonuniform Grids ◽  
Good Efficiency ◽  
Good Convergence ◽  
Implicit Difference Methods ◽  
Difference Methods ◽  
Grid Points ◽  
High Flexibility

A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.

Stability of Implicit Difference Equations Generated by Parabolic Functional Differential Problems

2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Numerical Examples ◽  
Convergence Results ◽  
Nonlinear Parabolic ◽  
Implicit Difference Methods ◽  
Neumann Type ◽  
Difference Methods ◽  
Functional Differential

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.

Explicit and Implicit Difference Methods for Quasilinear First Order Partial Functional Differential Equations

2014 ◽  
Vol 14 (2) ◽  
pp. 151-175
Author(s):  
Zdzisław Kamont ◽  
Anna Szafrańska
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Difference Schemes ◽  
Implicit Methods ◽  
Implicit Difference Methods ◽  
Difference Methods ◽  
Functional Differential ◽  
Explicit Difference Schemes

Abstract. Initial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes of the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. Error estimates for both methods are presented. Interpolating operators corresponding to functional variables are constructed.

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