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- Engblom, Stefan, et al.
(författare)
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Simulation of stochastic reaction-diffusion processes on unstructured meshes
- 2008
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Stochastic chemical systems with diffusion are modeled with 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 Brownian motion in space to obtain the reaction-diffusion 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.
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| 4. |
- Ferm, Lars, et al.
(författare)
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A hierarchy of approximations of the master equation scaled by a size parameter
- 2007
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Rapport (övrigt vetenskapligt/konstnärligt)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.
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| 7. |
- Ferm, Lars, et al.
(författare)
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Accurate and stable grid interfaces for finite volume methods
- 2002
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A convection-diffusion equation is discretized by a finite volume method in two space dimensions. The grid is partitioned into blocks with jumps in the grid size at the block interfaces. Interpolation in the cells adjacent to the interfaces is necessary to be able to apply the difference stencils. Second order accuracy is achieved and the stability of the discretizations is investigated. The interface treatment is tested in the solution of the compressible Navier-Stokes equations. The conclusions from the scalar equation are valid also for these equations.
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| 8. |
- Ferm, Lars, et al.
(författare)
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Adaptive, Conservative Solution of the Fokker-Planck Equation in Molecular Biology
- 2004
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The Fokker-Planck equation on conservation form is discretized by a finite volume method 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. Comparison is made with a stochastic method.
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| 10. |
- Ferm, Lars, et al.
(författare)
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Adaptive error control for steady state solutions of inviscid flow
- 2000
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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