Could someone suggest methods/algorithms to solve a system like
y' = My + f
where M is a huge matrix? Most explicit methods (like usual RK4) require very tiny timesteps for stiff systems, implicit methods are too slow and resource-consuming because of the size of M.
I'm interested also in (free, online available) review papers to summarize the progress in this field.
Some papers provide one-step discretization (Euler method). For example, see ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6588290 and file.scirp.org/pdf/AM20100300009_14716417.pdf Also, there are methods for solving large scale differential and differentiak-algebraic systems, which can be found in https://computation.llnl.gov/casc/nsde/pubs/221215.pdf (and see refferences therein). Also, if M is real natrix, the matrix exponential computation can be used. The system turns to its discrete-time analog of the form y(k+1)=exp(M*h)*y(k) +N*f(k), where $N = \int_{0}^{h}{exp(M*(t-h))dt}. How to compute matrix exponential, see http://www.cs.cornell.edu/cv/ResearchPDF/19ways+.pdf
Pseudo-spectral methods are the most efficient; finite difference methods have discretization errors that decay as powers of the integration step, while pseudo-spectral methods can achieve errors exponentially small in the time step. And they're more efficient in taking care of global errors.
Do M and/or f depend on y(t) and/or t? Assuming y' = dy/dt...
What are the initial conditions?
The answer to your question will depend on such information.
M and f can depend on y only slightly, so can be considered constant in a few time step. Also, M is not tridiagonal, but can be sparse, but not in a regular way.
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