Preconditioning waveform relaxation iterations for differential systems

We discuss preconditioning and overlapping of waveform relaxation methods for sparse linear differential systems. It is demonstrated that these techniques significantly improve the speed of convergence of the waveform relaxation iterations resulting from application of various modes of block Gauss-Jacobi and block Gauss-Seidel methods to differential systems. Numerical results are presented for linear systems resulting from semi-discretization of the heat equation in one and two space variables. It turns out that overlapping is very effective for the system corresponding to the one-dimensional heat equation and preconditioning is very effective for the system corresponding to the two-dimensional case