The solution of implicit differential equations on parallel computers Original Research Article

We construct and analyze parallel iterative solvers for the solution of the linear systems arising in the application of Newtonʹs method to s-stage implicit Runge-Kutta (RK) type discretizations of implicit differential equations (IDEs). These linear solvers are partly iterative and partly direct. Each linear system iteration again requires the solution of linear subsystems, but now only of IDE dimension, which is s times less than the dimension of the linear system in Newtonʹs method. Thus, the effective costs on a parallel computer system are only one LU-decomposition of IDE dimension for each Jacobian update, yielding a considerable reduction of the effective LU-costs. The method parameters can be chosen such that only a few iterations by the linear solver are needed. The algorithmic properties are illustrated by solving the transistor problem (index 1) and the car axis problem (index 3) taken from the CWI test set.