What Is DT?

What are the advantages of using explicit method (e.g. Forward Euler) over implicit methods, when the differential equations system likes this, Cdx(t)/dt=Gx(t)+u(t)Cdx(t)/dt=Gx(t)+u(t)C dx(t)/dt = G x(t)+u(t) (where CC C and GG G is larger sparse matrices, u(t)u(t)u(t) is input to this system, x(t)x

• If dx(t)/dt=Gx(t)+u(t)dx(t)/dt=Gx(t)+u(t) dx(t) /dt = Gx(t) + u(t) , it is simple and explicit method is good, only require matrix-vector multiplications, but how about when there is C matrix on the left hand side?

• Answer:

  It would depend on your matrix CC C . Systems like the one you give actually do arise frequently from finite element analysis, in those cases the matrix CCC is a mass matrix, and is therefore very well conditioned, symmetric, positive definite, the works. Therefore solves involving CC C are about as cheap as a large linear system can get, and if you stick to an explicit type method those are the only solves you will need to do (plus matrix-vector products with GGG). If you want to use an implicit method though, your solve will involve both CCC and GGG, this could be much more expensive to solve. It ends up being a perturbation of CCC by a multiple of GGG, and this might break positive definiteness,symmetry, etc, and the resulting system could be very poorly conditioned if that matters for the purpose of iterative solvers (though if you go down the iterative route in this case you are basically just doing an explicit method)