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Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

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Abstract

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, it helps conserving the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit.

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Acknowledgements

This work was partially supported by AFOSR under contract FA9550-17-1-0241.

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Authors and Affiliations

  1. Institute of Mathematics (MATH), School of Basic Sciences (FSB), EPFL, 1015, Lausanne, Switzerland

    Babak Maboudi Afkham & Jan S. Hesthaven

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  1. Babak Maboudi Afkham
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  2. Jan S. Hesthaven
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Correspondence to Babak Maboudi Afkham.

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Maboudi Afkham, B., Hesthaven, J.S. Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems. J Sci Comput 81, 3–21 (2019). https://doi.org/10.1007/s10915-018-0653-6

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  • DOI: https://doi.org/10.1007/s10915-018-0653-6

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