Parallel numerical methods for large-scale DAE systems

For plantwide dynamic simulation in chemical process industry, parallel numerical methods using a divide and conquer strategy are considered. An approach for the numerical solution of initial value problems for large systems of differential algebraic equations (DAEs) arising from industrial applications and its realization on parallel computers with shared memory is discussed. The system is partitioned into blocks and then it is extended appropriately, such that block-structured Newton-type methods can be applied which enable the application of relaxation techniques. This approach has gained considerable speedup factors for the dynamic simulation of various large-scale distillation plants, covering systems with up to 60 000 equations

SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics

We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22,380 nodes using 671,400 threads.Comment: 41 pages, 13 figures, and 7 tables. Ancillary data contains simulation input file

Enhancing speed and scalability of the ParFlow simulation code

Regional hydrology studies are often supported by high resolution simulations of subsurface flow that require expensive and extensive computations. Efficient usage of the latest high performance parallel computing systems becomes a necessity. The simulation software ParFlow has been demonstrated to meet this requirement and shown to have excellent solver scalability for up to 16,384 processes. In the present work we show that the code requires further enhancements in order to fully take advantage of current petascale machines. We identify ParFlow's way of parallelization of the computational mesh as a central bottleneck. We propose to reorganize this subsystem using fast mesh partition algorithms provided by the parallel adaptive mesh refinement library p4est. We realize this in a minimally invasive manner by modifying selected parts of the code to reinterpret the existing mesh data structures. We evaluate the scaling performance of the modified version of ParFlow, demonstrating good weak and strong scaling up to 458k cores of the Juqueen supercomputer, and test an example application at large scale.Comment: The final publication is available at link.springer.co

Waveform Relaxation for the Computational Homogenization of Multiscale Magnetoquasistatic Problems

This paper proposes the application of the waveform relaxation method to the homogenization of multiscale magnetoquasistatic problems. In the monolithic heterogeneous multiscale method, the nonlinear macroscale problem is solved using the Newton--Raphson scheme. The resolution of many mesoscale problems per Gauss point allows to compute the homogenized constitutive law and its derivative by finite differences. In the proposed approach, the macroscale problem and the mesoscale problems are weakly coupled and solved separately using the finite element method on time intervals for several waveform relaxation iterations. The exchange of information between both problems is still carried out using the heterogeneous multiscale method. However, the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per Gauss point.Comment: submitted to JC

Reducing phase error in long numerical binary black hole evolutions with sixth order finite differencing

We describe a modification of a fourth-order accurate ``moving puncture'' evolution code, where by replacing spatial fourth-order accurate differencing operators in the bulk of the grid by a specific choice of sixth-order accurate stencils we gain significant improvements in accuracy. We illustrate the performance of the modified algorithm with an equal-mass simulation covering nine orbits.Comment: 13 pages, 6 figure

A task-based approach to parallel parametric linear programming solving, and application to polyhedral computations

Parametric linear programming is a central operation for polyhedral computations, as well as in certain control applications.Here we propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems.This type of parallel applications is challenging, because several tasks mightbe computing the same region. In this paper, we are presenting thealgorithm itself with a parallel redundancy elimination algorithm, andconducting a thorough performance analysis.Comment: arXiv admin note: text overlap with arXiv:1904.0607

Large-Scale Structure in Brane-Induced Gravity II. Numerical Simulations

We use N-body simulations to study the nonlinear structure formation in brane-induced gravity, developing a new method that requires alternate use of Fast Fourier Transforms and relaxation. This enables us to compute the nonlinear matter power spectrum and bispectrum, the halo mass function, and the halo bias. From the simulation results, we confirm the expectations based on analytic arguments that the Vainshtein mechanism does operate as anticipated, with the density power spectrum approaching that of standard gravity within a modified background evolution in the nonlinear regime. The transition is very broad and there is no well defined Vainshtein scale, but roughly this corresponds to k_*~ 2 at redshift z=1 and k_*~ 1 at z=0. We checked that while extrinsic curvature fluctuations go nonlinear, and the dynamics of the brane-bending mode C receives important nonlinear corrections, this mode does get suppressed compared to density perturbations, effectively decoupling from the standard gravity sector. At the same time, there is no violation of the weak field limit for metric perturbations associated with C. We find good agreement between our measurements and the predictions for the nonlinear power spectrum presented in paper I, that rely on a renormalization of the linear spectrum due to nonlinearities in the modified gravity sector. A similar prediction for the mass function shows the right trends. Our simulations also confirm the induced change in the bispectrum configuration dependence predicted in paper I.Comment: 19 pages, 13 figures. v2: corrected typos, added more simulations, better test of predictions in large mass regime. v3: minor changes, published versio