Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian


  • Shev Macnamara University of Technology Sydney
  • Sergio Blanes Instituto de Matemática Multidisciplinar, Universitat Politècnica de València
  • Arieh Iserles Department of Applied Mathematics and Mathematical Physics, University of Cambridge



graph laplacian, pseudospectra, magnus expansion,


An important framework for modelling and simulation of chemical reactions is a Markov process sometimes known as a master equation. Explicit solutions of master equations are rare; in general the explicit solution of the governing master equation for a bimolecular reaction remains an open question. We show that a solution is possible in special cases. One method of solution is diagonalization. The crucial class of matrices that describe this family of models are non-symmetric graph Laplacians. We illustrate how standard numerical algorithms for finding eigenvalues fail for the non-symmetric graph Laplacians that arise in master equations for models of chemical kinetics. We propose a novel way to explore the pseudospectra of the non-symmetric graph Laplacians that arise in this class of applications, and illustrate our proposal by Monte Carlo. Finally, we apply the Magnus expansion, which provides a method of simulation when rates change in time. Again the graph Laplacian structure presents some unique issues: standard numerical methods of more than second-order fail to preserve positivity. We therefore propose a method that achieves fourth-order accuracy, and maintain positivity.


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2020-06-16 — Updated on 2020-06-16



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