80A30 Chemical kinetics [See also 76V05, 92C45, 92E20]
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Keywords
- chemical master equation (2)
- finite volume method (2)
- heterostructures (2)
- reaction-diffusion systems (2)
- convergence (1)
- discrete Gagliardo-Nirenberg inequalities (1)
- discrete Moser iteration (1)
- error of finite state projections (1)
- existence of solutions and moments (1)
- finite state projections (1)
The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted $\ell_1$-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.
We investigate the convergence of an implicit Voronoi finite volume
method for reaction-diffusion problems including nonlinear diffusion
in two space dimensions. The model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as
primary variables. The numerical scheme uses boundary conforming Delaunay
meshes and preserves positivity and the dissipative property of the
continuous system. Starting from a result on the global stability of
the scheme (uniform, mesh-independent global upper and lower bounds),
we prove strong convergence of the chemical activities and their gradients
to a weak solution of the continuous problem. In order to illustrate the
preservation of qualitative properties by the numerical scheme, we present
a long-term simulation of the Michaelis-Menten-Henri system. Especially,
we investigate the decay properties of the relative free energy and the
evolution of the dissipation rate over several magnitudes of time, and
obtain experimental orders of convergence for these quantities.
The chemical master equation is a fundamental equation in chemical kinetics. It is an adequate substitute for the classical reaction-rate equations whenever stochastic effects become relevant. In the present paper we give a simple argument showing that the solutions of a large class of chemical master equations, including all those in which elementary reactions between two and more molecules do not generate a larger number of molecules than existed before,
are bounded in weighted $\ell_1$-spaces. As an illustration for the implications of this kind of regularity we analyze the effect of truncating the state space. This leads to an error analysis of the finite state projection of the chemical master equation, an approximation that underlies many numerical methods.
We consider discretizations for reaction-diffusion systems with nonlinear
diffusion in two space dimensions. The applied model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as primary variables.
We propose an implicit Voronoi finite volume discretization on regular Delaunay
meshes that allows to prove uniform, mesh-independent global upper and lower L1
bounds for the chemical potentials. These bounds provide the main step for a convergence
analysis for the full discretized nonlinear evolution problem. The fundamental
ideas are energy estimates, a discrete Moser iteration and the use of discrete
Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities
we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides
with the gradient norm of continuous piecewise linear finite elements.