Conservative solution of the Fokker–Planck equation for stochastic chemical reactions

The Fokker–Planck equation on conservation form modeling stochastic chemical reactions is discretized by a finite volume method for low dimensional problems and advanced in time by a linear multistep method. The grid cells are refined and coarsened in blocks of the grid depending on an estimate of the spatial discretization error and the time step is chosen to satisfy a tolerance on the temporal discretization error. The solution is conserved across the block boundaries so that the total probability is constant. A similar effect is achieved by rescaling the solution. The steady state solution is determined as the eigenvector corresponding to the zero eigenvalue. The method is applied to the solution of a problem with two molecular species and the simulation of a circadian clock in a biological cell. Comparison is made with a Monte Carlo method.

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