A multidomain spectral method for solving elliptic equations

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.Comment: 31 pages, 8 figure

Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace methods.Comment: 15 pages, 5 figures, 1 table, 2 algorithm

Afterlive: A performant code for Vlasov-Hybrid simulations

A parallelized implementation of the Vlasov-Hybrid method [Nunn, 1993] is presented. This method is a hybrid between a gridded Eulerian description and Lagrangian meta-particles. Unlike the Particle-in-Cell method [Dawson, 1983] which simply adds up the contribution of meta-particles, this method does a reconstruction of the distribution function $f$ in every time step for each species. This interpolation method combines meta-particles with different weights in such a way that particles with large weight do not drown out particles that represent small contributions to the phase space density. These core properties allow the use of a much larger range of macro factors and can thus represent a much larger dynamic range in phase space density. The reconstructed phase space density $f$ is used to calculate momenta of the distribution function such as the charge density $\rho$. The charge density $\rho$ is also used as input into a spectral solver that calculates the self-consistent electrostatic field which is used to update the particles for the next time-step. Afterlive (A Fourier-based Tool in the Electrostatic limit for the Rapid Low-noise Integration of the Vlasov Equation) is fully parallelized using MPI and writes output using parallel HDF5. The input to the simulation is read from a JSON description that sets the initial particle distributions as well as domain size and discretization constraints. The implementation presented here is intentionally limited to one spatial dimension and resolves one or three dimensions in velocity space. Additional spatial dimensions can be added in a straight forward way, but make runs computationally even more costly.Comment: Accepted for publication in Computer Physics Communication