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Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws

  • Willy Hereman
  • Michael Colagrosso
  • Ryan Sayers
  • Adam Ringler
  • Bernard Deconinck
  • Michael Nivala
  • Mark Hickman
Part of the Trends in Mathematics book series (TM)

Abstract

We introduce calculus-based formulas for the continuous Euler and homotopy operators. The 1D continuous homotopy operator automates integration by parts on the jet space. Its 3D generalization allows one to invert the total divergence operator. As a practical application, we show how the operators can be used to symbolically compute local conservation laws of nonlinear systems of partial differential equations in multi-dimensions.

Analogous to the continuous case, we also present concrete formulas for the discrete Euler and homotopy operators. Essentially, the discrete homotopy operator carries out summation by parts. We use it to algorithmically invert the forward difference operator. We apply the discrete operator to compute fluxes of differential-difference equations in (1 + 1) dimensions.

Our calculus-based approach allows for a straightforward implementation of the operators in major computer algebra systems, such as Mathematica and Maple. The symbolic algorithms for integration and summation by parts are illustrated with elementary examples. The algorithms to compute conservation laws are illustrated with nonlinear PDEs and their discretizations arising in fluid dynamics and mathematical physics.

Keywords

Homotopy operator conservation law integrability testing 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Willy Hereman
    • 1
  • Michael Colagrosso
    • 1
  • Ryan Sayers
    • 1
  • Adam Ringler
    • 1
  • Bernard Deconinck
    • 2
  • Michael Nivala
    • 2
  • Mark Hickman
    • 3
  1. 1.Department of Mathematical and Computer SciencesColorado School of MinesGoldenUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

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