Stochastic chemical systems with diffusion are modeled with a reactiondiffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with Brownian motion in space to obtain the reactiondiffusion master equation. The space is covered by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso-or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology.

We propose an adaptive hybrid method suitable for stochastic simulation of diffusion dominated reaction-diffusion processes. For such systems, simulation of the diffusion requires the predominant part of the computing time. In order to reduce the computational work, the diffusion in parts of the domain is treated macroscopically, in other parts with the tau-leap method and in the remaining parts with Gillespie's stochastic simulation algorithm (SSA) as implemented in the next subvolume method (NSM). The chemical reactions are handled by SSA everywhere in the computational domain. A trajectory of the process is advanced in time by an operator splitting technique and the time steps are chosen adaptively. The spatial adaptation is based on estimates of the errors in the tau-leap method and the macroscopic diffusion. The accuracy and efficiency of the method are demonstrated in examples from molecular biology where the domain is discretized by unstructured meshes.

Solutions of the master equation are approximated using a hierarchy of models based on the solution of ordinary differential equations: the macroscopic equations, the linear noise approximation and the moment equations. The advantage with the approximations is that the computational work with deterministic algorithms grows as a polynomial in the number of species instead of an exponential growth with conventional methods for the master equation. The relation between the approximations is investigated theoretically and in numerical examples. The solutions converge to the macroscopic equations when a parameter measuring the size of the system grows. A computational criterion is suggested for estimating the accuracy of the approximations. The numerical examples are models for the migration of people, in population dynamics and in molecular biology.

The Fokker-Planck equation models chemical reactions on a mesoscale. The solution is a probability density function for the copy number of the different molecules. The number of dimensions of the problem can be large making numerical simulation of the reactions computationally intractable. The number of dimensions is reduced here by deriving partial differential equations for the first moments of some of the species and coupling them to a Fokker-Planck equation for the remaining species. With more simplifying assumptions, another system of equations is derived consisting of integro-differential equations and a Fokker-Planck equation. In this way, the simulation of the chemical networks is possible without the exponential growth in computatational work and memory of the original equation and with better modelling accuracy than the macroscopic reaction rate equations. Some terms in the equations are small and are ignored. Conditions are given for the influence of these terms to be small on the equations and the solutions. The difference between different models is illustrated in a numerical example.

The steady state solution of the Euler equations of inviscid flow is computed by an adaptive method. The grid is structured and is refined and coarsened in predefined blocks. The equations are discretized by a finite volume method. Error equations, satisfied by the solution errors, are derived with the discretization error as the driving right-hand side. An algorithm based on the error equations is developed for errors propagated along streamlines. Numerical examples from two-dimensional compressible and incompressible flow illustrate the method.Key words. finite volume method, discretization error, error control, error equation, Euler equations AMS subject classifications. 65N15, 65N50 PII. S1064827500367452 Introduction. Inadequate grid resolution is the most important contributor to inaccuracy in computational fluid dynamics (CFD) calculations [31]. This is a serious obstacle in the industrial use of CFD methods. Grid refinement studies reveal the sensitivity of the solution to the grid spacing. Such studies are expensive and timeconsuming to make for an applications engineer, and usually only one grid is generated based on his or her experience. There is a need for solution-adaptive methods.The errors in the numerical solution of the partial differential equations (PDEs) governing inviscid flow can be controlled by an adaptive method. By changing the grid spacing so that the solution errors fulfill given tolerances, the quality of the solution is guaranteed. Furthermore, computational work and memory is saved since the grid is not made unnecessarily fine. Two important ingredients in an adaptive algorithm are a way of estimating the solution error and a data structure for the grid that allows for modifications of the original grid. We combine computing the solution errors from the error equations with estimated discretization errors as source terms with refinement and coarsening in predetermined blocks of the grid.The equations are discretized by finite volume methods in this paper. They are of second order on Cartesian grids. The refinement and the coarsening of the grid is determined by estimates of the solution error. Grid adaptation is necessary not only at discontinuities in the solution but also in the smooth parts. Differential equations are derived which are satisfied approximately by the errors. The right-hand side of the error equations is the discretization error in the finite volume method. This error is estimated by comparing the space discretization on two different grids. Then the error equations are solved numerically for the error in the solution. For certain variables, these equations are particularly simple and the error is propagated along the streamlines of the flow. An algorithm is devised for such errors. It is relatively easily implemented in existing codes and takes the sign of the error into account when the grid is adjusted so that a tolerance on the maximum error is fulfilled. The

An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge-Kutta-Fehlberg method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The error equation is integrated to obtain global errors of the solution. The method is shown to be stable if one-sided space discretizations are used. Examples such as the wave equation, Burgers' equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.

Dedicated to Björn Engquist on the occasion of his 60th birthday. Abstract.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. (2000): 65M20, 65M50. AMS subject classification

The master equation satisfied by a probability density function is solved on a grid with a cell size 1 A modified master equation is derived for the time development of the average of the density in the larger cells. The accuracy of the approximation is studied and the total probability is conserved. Based on an estimate of the discretization error, the cell size is dynamically adapted to the solution. The method is suitable for a few space dimensions and is tested on a model for the migration of people. Substantial savings in memory requirements and CPU times are reported in numerical experiments.