Parallel exponential Rosenbrock methods

Exponential Rosenbrock integrators were shown to be very efficient in solving large systems of stiff ordinary differential equations. So far, such exponential methods have been derived up to order 5. The aim of this paper is to construct new integrators of orders 4, 5, and 6. In contrast to the existing schemes, the new schemes, which are called parallel exponential Rosenbrock integrators, can be implemented on a multi-processor system or parallel computers. The new schemes of orders 4 and 5 require the same number of stages as the old schemes of the same orders of accuracy. However, while the parallel integrator of order 4 can be implemented with the same cost as a 2-stage method, the ones of orders 5 and 6 can be implemented at the cost of a 3-stage method only. This offers a significant improvement over the old schemes in terms of computational time when implemented in parallel. The numerical experiments show the efficiency of the new integrators as well as the comparative performance with the old ones.