Is there any explicit methods to solve ''stiff'' coupled differential equations?

Question

I am trying to solve coupled ordinary,partial differential equations, reaction diffusion equations with finite difference method.
But I found that these equations are 'stiff'. I searched that when I want to solve 'stiff' problems, I have to use an implicit method. But there is one problem. The constant in my equations changes according to the previous value of $y$. ($y$ is the value I want to calculate) So I cannot use an implicit method because the constant in equations has to be decided from the previous step. If there is anyone who recommend some methods to solve this problem, Please inform me of that.
Thank you.

Answer 1

The DASSL algorithm may be suited to your needs (http://www.osti.gov/scitech/servlets/purl/5882821), it is well tested and used in commercial dynamic modelling packages such as Dymola.

it is best suited to stiff implicitly coupled systems of Differential Algebraic Equations of the form.

$$F(y(t),\dot{y(t)} ,x(t), t ) =0 $$

Where $y$, $\dot{y}$ are vectors of time varying variables and their first order time derivative, and $x$ is a vector of variables which do not appear with a corresponding time derivative. It is applies to cases where there is no $x$ (a pure ODE system).

Note, not to be confused between uses of the word 'implicit'. The algorithm is explicit in that it only requires knowledge of the previous time interval. But it solves an implicitly defined system.

There are many other options too, I recommend the book 'Continuous System Simulation' by Francois Cellier and Ernesto Kofman for a detailed overview of numerical methods for stiff systems, PDEs, ODEs and large system modelling.

Answer 2

A particularly well suited explicit method you can use is Exponential Integration based on Krylov subspace projections. The method has succesfully been used to solve reaction diffusion problems cheaply, as this method does not have the strict time-step restrictions that other explicit methods show due to the stiffness of the equations.

You can find a comprehensive description about its implementation and derivation in the following publications:

• Saad, Y. (1992). Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM Jounral of Numerical Analysis, 209-228.

• Schulze, J. C., Schmid, P. J., & Sesterhenn, J. L. (2009). Exponential time integration using Krylov subspaces. International Journal for Numerical Methods in Fluids, 591-609.