Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian
[EN] 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 quest...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Recursos: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:riunet.upv.es:10251/163485 |
| Acesso em linha: | https://riunet.upv.es/handle/10251/163485 |
| Access Level: | acceso abierto |
| Palavra-chave: | Graph laplacian Pseudospectra Magnus expansion MATEMATICA APLICADA |
| Resumo: | [EN] 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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