Stefan Engblom
Division of Scientific Computing
Department of Information Technology
Uppsala University
The master equation of chemical reactions is an accurate stochastic
description of quite general systems in chemistry. For D reacting
species this is a differential-difference equation in D dimensions
suffering from the usual exponential solution complexity.
The most frequently used approximative solution method is to write
down the corresponding set of reaction rate equations. In many
cases this approximation is not valid, or only partially so, as
stochastic effects caused by the natural noise present in the full
description of the problem are poorly captured.
We will show how ``higher order'' reaction rate equations can be
derived governing the first few moments of the solution. As an example
we consider a chemical system describing a circadian clock and show
that these equations can produce solutions that better capture
stochastic effects while still maintaining the low complexity of the
reaction rate approach.
We will also consider a different angle of attack and present an
adaptive spectral method of low complexity. The method is unusual in
its choice of basis functions, its adaptivity and its relatively high
accuracy.