Some Theoretical Considerations

  • J. W. Thomas
Part of the Texts in Applied Mathematics book series (TAM, volume 22)


This chapter begins with the basic definitions of what it means for the solution of the finite difference equations to converge to the solution of the partial differential equation. It is extremely important for a user of finite difference techniques to understand precisely what type of convergence their scheme has, what kinds of assumptions are made to get this convergence and how this convergence affects their accuracy. The most common approach to convergence of difference equations is through the concepts of consistency and stability and the Lax Theorem. The Lax Theorem allows us to prove convergence of a difference scheme by showing that the scheme is both consistent and stable (which are generally easier to show). The goal of this chapter is to include these definitions and the Lax Theorem in a rigorous setting that is intuitive and understandable.


Partial Differential Equation Difference Scheme Difference Equation Neumann Boundary Condition Implicit Scheme 
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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • J. W. Thomas
    • 1
  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA

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