Pseudospectral solution of linear evolution equations of second order in space and time on unstructured quadrilateral subdomain topologies

A multidomain Legendre pseudospectral method is developed for the solution of linear hyperbolic initial boundary value problems, with mixed boundary conditions, in general two-dimensional and axisymmetric geometries. A weak collocation spectral method is utilized for the spatial approximation of a generic wave evolution equation over multiple nonoverlapping subdomains. The system of ordinary differential equations that stems from the above procedure is integrated in time by implicit as well as explicit high order temporal approximation algorithms. The weak formalism of the influence matrix method is combined with the implicit approximation, so as to efficiently solve the coupled system of linear equations after the full discretization, while a novel technique for avoiding the amplification of roundoff error at high temporal resolution simulations with the implicit temporal integration methods, is also studied. An innovative method for the treatment of Dirichlet boundary conditions is proposed, in order to avoid the order reduction which usually arises with the utilization of the explicit time integrator. Furthermore, appropriate modifications are reported, for dealing with the pole singularity problem faced by the weak formulation of axisymmetric problems. Finally, numerical simulations of a variety of wave problems on curvilinear geometries and unstructured subdomain configurations are presented in order to assess the capabilities of the proposed methodology in handling efficiently general hyperbolic differential operators.