Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

  • Authors:

  • Dirk Lebiedz;Volkmar Reinhardt;Jochen Siehr

  • Affiliations:

  • Center for Analysis of Biological Systems (ZBSA), University of Freiburg, Habsburgerstraíe 49, 79104 Freiburg, Germany and Interdisciplinary Center for Scientific Computing (IWR), University ...;Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany and SEW-EURODRIVE GmbH & Co KG, Ernst-Blickle-Str. 42, 76646 B ...;Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of ''minimal relaxation'' of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to four test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.