The model was solved numerically by using the finite element method and the orthogonal collocation method for the discretization of partial differential equations and Gear's stiff method for the solution of the resulting ordinary differential equation system.
The Rosenbrock (Stiff) method is used to integrate the differential equations with a tolerance of 1.0E-7.
The whole circadian clock pathway is modeled using a set of ODEs which were numerically solved by stiff method.
Our non-stiff method is expected to apply widely to problems for doubly-periodic interfacial flow with surface tension which have a boundary integral formulation.
The genesis of the PAIRS approach in [1] was the need for a non-stiff method to keep rigid bodies apart, i.e. resolution of collisions.
For the other molecules, numerical solutions of equation (3) were obtained with Mathematica® (Solver NDSolve: it automatically switches between non-stiff Adams method and stiff Gear method).
For integration in time three different codes are used, including standard stiff ODE methods.
For systems with severely stiff reactions, those methods are not efficient because the reaction terms in IF or ETD are still approximated with explicit schemes.
The introduction of detailed reaction mechanisms in multi-dimensional simulations is limited by unfavorable scaling of the stiff ODE solution methods with the mechanism's size.
When the reactions are stiff, existing numerical methods for stochastic reaction diffusion equations require either very small time steps for any explicit schemes or solving large nonlinear systems at each time step for the implicit schemes.
