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

Hyperbolic systems of conservation laws that are discretized in space by spectral collocation methods and advanced in time by finite difference schemes are considered. At any time-level a domain decomposition method is introduced that is 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.