Multidomain Spectral Method for the Helically Reduced Wave Equation

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the "eigenspectral method." Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.Comment: 57 pages, 11 figures, uses elsart.cls. Final version includes revisions based on referee reports and has two extra figure

A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method

A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct (as opposed to iterative) solver that has optimal O(N) complexity for all stages of the computation when applied to problems with non-oscillatory solutions such as the Laplace and the Stokes equations. Numerical examples demonstrate that the scheme is capable of computing solutions with relative accuracy of $10^{-10}$ or better, even for challenging problems such as highly oscillatory Helmholtz problems and convection-dominated convection diffusion equations. In terms of speed, it is demonstrated that a problem with a non-oscillatory solution that was discretized using $10^{8}$ nodes was solved in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since the solver is direct, and the "solution operator" fits in RAM, any solves beyond the first are very fast. In the example with $10^{8}$ unknowns, solves require only 30 seconds.Comment: arXiv admin note: text overlap with arXiv:1302.599

SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs

The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two dimensional domain. The solver is designed to reduce communication costs and perform well on GPUs; it uses a two-level framework, which is easier to implement and optimize than traditional multi-frontal schemes based on hierarchical nested dissection orderings. The scheme decomposes the domain into thin subdomains, or "slabs". Within each slab, a local factorization is executed that exploits the geometry of the local domain. A global factorization is then obtained through the LU factorization of a block-tridiagonal reduced coefficient matrix. The solver has complexity $O(N^{5/3})$ for the factorization step, and $O(N^{7/6})$ for each solve once the factorization is completed. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order convergent multi-domain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size $1000 \lambda \times 1000 \lambda$ (for which N=100 \mbox{M}) is solved in 15 minutes to 6 correct digits on a high-powered desktop with GPU acceleration