Difference Methods for Parabolic History Value Problems.

Abstract

This paper is concerned with one-step difference methods for parabolic history value problems in one space variable. These problems, which have the feature that the evolution of the solution is influenced by 'all its past' occur in the theory of viscoelastic liquids (materials with 'memory'). The history dependence is represented by a Volterra-integral in the equation of motion. Using recently obtained existence results and smoothness assumptions on the solution, we derive a local stability and convergence result for a Crank-Nicolson-type difference scheme by interpreting the linearized scheme as perturbation of a strictly parabolic scheme without memory term. Second order convergence is shown on sufficiently small time intervals. The presented approach carries over to other one-step difference methods like implicit Euler schemes.

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Document Details

Document Type
Technical Report
Publication Date
Mar 01, 1982
Accession Number
ADA114597

Entities

People

  • Peter Markowich

Organizations

  • University of Wisconsin–Madison

Tags

DTIC Thesaurus Topics

  • Banach Space
  • Boundary Value Problems
  • Computer Science
  • Computers
  • Convergence
  • Difference Equations
  • Differential Equations
  • Equations
  • Formulas (Mathematics)
  • Materials
  • Mathematics
  • Numerical Analysis
  • Perturbations
  • Smoothing (Mathematics)
  • Theorems
  • United States
  • Universities

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Theoretical Analysis.

Technology Areas

  • Space