Lars Ferm

Simulation of Stochastic Reaction-Diffusion Processes on Unstructured Meshes

We model stochastic chemical systems with diffusion by a reaction-diffusion 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 a discretized Brownian motion in space to obtain the reaction-diffusion master equation. The space is covered in our method 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.

An adaptive algorithm for simulation of stochastic reaction-diffusion processes

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 Gillespies 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 timesteps 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.

A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter

Abstract 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.

A down-stream boundary procedure for the euler equations

Abstract The steady slate problem for flow in a channel is considered. Down-stream boundary conditions are derived for the Euler equations by using Fourier-expansions of the solution. Not data are required at the down-stream boundary or at infinity, the condition used is simply the requirement of a bounded solution. A difference approximation is applied to the steady state equations, and Newtons method is used for solving the resulting non-linear system of equations.

Open boundary conditions for external flow problems

Abstract The steady Euler equations are considered. Very accurate open boundary conditions are derived for the external problem when the outer boundary is an ellipse. These conditions have the same algebraic form as the corresponding conditions when the boundary is a straight line across an infinitely long channel. A new implementation is introduced for the external problem. At every time step a matrix is applied on a vector containing values from every grid point at the boundary. The computational work for these calculations is kept low by introducing a special set of fewer boundary condition points. Experiments demonstrate the accuracy of the boundary procedure.

Dimensional Reduction of the Fokker–Planck Equation for Stochastic Chemical Reactions

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 with a Fokker–Planck equation for the remaining species. With more simplifying assumptions, another system of equations is derived consisting of integrodifferential equations and a Fokker–Planck equation. In this way, the simulation of the chemical networks is possible without the exponential growth in computational work and memory of the original equation and with better modeling 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 ...

Adaptive Error Control for Steady State Solutions of Inviscid Flow

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.

Space---Time Adaptive Solution of First Order PDES

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 (RKF) method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. 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.

Open boundary conditions for stationary inviscid flow problems

Abstract The steady state problem for flow in a channel is considered. Very accurate open upstream and downstream boundary conditions are derived for the Euler equations. No data are needed at the boundaries. It is enough that the upstream limit of the solution is available. Error estimates are derived for a simple model problem, and numerical experiments verify the results for the true non-linear equations. Newtons method is used to solve the equations. The boundary conditions are given in such a form that they also can be used in connection with a time-dependent procedure.

On numerical errors in the boundary conditions of the Euler equations

Numerical errors in solution of the Euler equations of fluid flow are studied. The error equations are solved to analyze the propagation of the discretization errors. In particular, the errors caused by the boundary conditions and their propagation are investigated. Errors generated at a wall are transported differently in subsonic and supersonic flows. Accuracy may be lost due to the accumulation of errors along the walls. This can be avoided by increasing the accuracy of the boundary conditions. Large errors may still arise locally at the leading edge of a wing profile. There, a fine grid is the best way to reduce the error.