I wonder: what is the best algorithm to solve \begin{equation}\frac{du}{dt} = Au\end{equation}Where $A$ is a real $n\times n$ matrix. A is not explicitly time dependent, usually sparse but not necessarily banded.  It's eigenvalues have non-positive real parts.  A is also diagonalizable but may be too big for a full diagonalization to be computationally efficient.There is the implicit Trapezoidal rule which I have had a good experience. \begin{equation}(I-\frac{\Delta t}{2} A) u_{n+1} = (I+\frac{\Delta t}{2} A) u_{n}\end{equation}What about explicit methods or Pade approximants? Also, how does this changes if a forcing term is added to the RHS?
Algorithms for linear system of ODEs
linear algebra;ode
As your matrix is independent of $u$ the result is a matrix exponential times the intial vector. The standard discussion of relevent method can be found from http://scholar.google.at by searching for ''Nineteen dubious ways''.For the scaling-and-squaring algorithm (the least dubious one), see alsohttp://blogs.mathworks.com/cleve/2012/07/23/a-balancing-act-for-the-matrix-exponential/