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Differential Equations with Symbolic Computation pp 255–290Cite as

Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws

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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

In memory of Ryan Sayers (1982–2003)

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Author information

Authors and Affiliations

  1. Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO, 80401-1887, USA

    Willy Hereman, Michael Colagrosso, Ryan Sayers & Adam Ringler

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195-2420, USA

    Bernard Deconinck & Michael Nivala

  3. Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

    Mark Hickman

Authors
  1. Willy Hereman
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  2. Michael Colagrosso
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  3. Ryan Sayers
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  4. Adam Ringler
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  5. Bernard Deconinck
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  6. Michael Nivala
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  7. Mark Hickman
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Editor information

Editors and Affiliations

  1. School of Science, Beihang University, 37 Xueyuan Road, Beijing, 100083, China

    Dongming Wang & Zhiming Zheng & 

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie - CNRS, 8, rue due Capitaine Scott, 75015, Paris, France

    Dongming Wang

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Hereman, W. et al. (2005). Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws. In: Wang, D., Zheng, Z. (eds) Differential Equations with Symbolic Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7429-2_15

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