Spectral methods for hyperbolic initial boundary value problems on parallel computers

Abstract

In this paper a parallel spectral algorithm is developed for hyperbolic initial boundary value problems in one space dimension. The Galerkin-Collocation method, which is spectrally accurate in both space and time, is parallelized by using domain decomposition. This procedure leads to a minimization problem in which there is coupling at inter-domain boundaries. We construct a decoupled preconditioner which can be used to iteratively solve the minimization problem. Symmetric formulation of the problem, which is needed to compute the residual for the normal equations, is discussed. The methodology outlined for computing the normal equations applies equally well to computation of the residual for the p and h– p versions of the finite element method. There is, therefore, no need to compute the mass and stiffness matrices to obtain the residual, as is normally done. This leads to a great saving in time and memory particularly for solving nonlinear problems using the p and h– p versions of the finite element method. The method we discuss in this paper generalizes to hyperbolic initial boundary value problems in multidimensions too, provided the computational boundaries we have introduced are noncharacteristic and the system is symmetrizable. Finally, we show that for the case of analytic coefficients and data, satisfying all the required compatibility conditions so that the solution is analytic, the numerical solution is exponentially accurate in N, where N is proportional to the number of subdomains and the number of degrees of freedom in each element.