Difference Methods for One-Dimensional PDE

  • Simon Širca
  • Martin Horvat
Part of the Graduate Texts in Physics book series (GTP)

Abstract

Finite-difference methods for one-dimensional partial differential equations are introduced by first identifying the classes of equations upon which suitable discretizations are constructed. It is shown how parabolic equations and the corresponding boundary conditions are discretized such that a desired local order of error is achieved and that the discretization is consistent and yields a stable and convergent solution scheme. Convergence criteria are established for a variety of explicit and implicit difference schemes. Energy estimates and theorems on maxima are given as auxiliary tools that allow us to ascertain that the solutions are physically meaningful. Difference schemes for hyperbolic equations are introduced from the standpoint of the Courant–Friedrich–Lewy criterion, dispersion and dissipation. Various techniques for non-linear equations and equations of mixed type are given, including high-resolution schemes for equations that can be expressed in terms of conservation laws. The Problems include the (parabolic) diffusion and (hyperbolic) advection equation, Burgers equation, the shock-tube problem, Korteweg–de Vries equation, and the non-stationary linear and cubic Schrödinger equations.

Keywords

Difference Scheme Implicit Scheme Partial Differential Equation Nicolson Scheme Homogeneous Dirichlet Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Simon Širca
    • 1
  • Martin Horvat
    • 1
  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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