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SPATIALLY ADAPTIVE STOCHASTIC NUMERICAL METHODS FOR INTRINSIC FLUCTUATIONS IN REACTION-DIFFUSION
 

Summary: SPATIALLY ADAPTIVE STOCHASTIC NUMERICAL METHODS
FOR INTRINSIC FLUCTUATIONS IN REACTION-DIFFUSION
SYSTEMS
PAUL J. ATZBERGER
Abstract. Stochastic partial differential equations are introduced for the continuum concen-
tration fields of reaction-diffusion systems. The stochastic partial differential equations account for
fluctuations arising from the finite number of molecules which diffusively migrate and react. Spa-
tially adaptive stochastic numerical methods are developed for approximation of the stochastic partial
differential equations. The methods allow for adaptive meshes with multiple levels of resolution, Neu-
mann and Dirichlet boundary conditions, and domains having geometries with curved boundaries. A
key issue addressed by the methods is the formulation of consistent discretizations for the stochastic
driving fields at coarse-refined interfaces of the mesh and at boundaries. Methods are also introduced
for the efficient generation of the required stochastic driving fields on such meshes. As a demon-
stration of the methods, investigations are made of the role of fluctuations in a biological model for
microorganism direction sensing based on concentration gradients. Also investigated, a mechanism
for spatial pattern formation induced by fluctuations. The discretization approaches introduced for
SPDEs have the potential to be widely applicable in the development of numerical methods for the
study of spatially extended stochastic systems.
Key words. Adaptive Methods, Stochastic Numerical Methods, Stochastic Partial Differ-
ential Equations, Reaction-Diffusion, Multilevel Meshes, MAC Discretization, Statistical Physics,

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics