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Journal of Scientific Computing

, Volume 16, Issue 1, pp 47–67 | Cite as

Adaptive High-Order Finite-Difference Method for Nonlinear Wave Problems

  • I. Fatkullin
  • J. S. Hesthaven
Article

Abstract

We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity.

high-order finite-difference wavelets adaptivity nonlinear optics 

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Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • I. Fatkullin
    • 1
  • J. S. Hesthaven
    • 2
  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroy
  2. 2.Division of Applied MathematicsBrown UniversityProvidence, Rhode Island

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