Domain Decomposition Methods for Systems of Conservation Laws: Spectral Collocation Approximations

We consider hyperbolic systems of conservation laws which are discretized in space by spectral collocation methods and advanced in time by finite difference schemes. At any time-level we introduce a domain decomposition method based on an iteration-by-subdomain procedure yielding at each step a sequence of independent subproblems (one for each subdomain) that can be solved simultaneously. The method is set for a general nonlinear problem in several space variables. The convergence analysis, however, is carried out only for a linear one-dimensional system with continuous solutions. A precise form of the error reduction factor at each iteration is derived. Although the method is applied here to the case of spectral collocation approximation only, the idea is fairly general and can be used in a different context as well. For instance, its application to space discretization by finite differences is straightforward.