Spectral approximation of solutions to the chemical master equation

The master equation of chemical reactions is an accurate stochastic description of general systems in chemistry. For DD reacting species this is a differential-difference equation in DD dimensions, exactly soluble for very simple systems only.We propose and analyze a novel solution strategy in the form of a Galerkin spectral method with a favorable choice of basis functions. A spectral approximation theory in the corresponding spaces is developed and the issue of stability is discussed.The convergence properties of the method are demonstrated by the numerical solution of two model problems with known solutions and a third problem for which no solution is known. It is shown that the method is effective and accurate, providing a viable alternative to other solution methods when the dimensionality is not too high.