Domain decomposition methods for the parallel computation of reacting flows

Domain decomposition is a natural route to parallel computing for partial differential equation solvers. Subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, comparisons are made between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demonstrate for it approximately 10-fold speedup on 16 processors

Analysis of a parallelized nonlinear elliptic boundary value problem solver with application to reacting flows

A parallelized finite difference code based on the Newton method for systems of nonlinear elliptic boundary value problems in two dimensions is analyzed in terms of computational complexity and parallel efficiency. An approximate cost function depending on 15 dimensionless parameters is derived for algorithms based on stripwise and boxwise decompositions of the domain and a one-to-one assignment of the strip or box subdomains to processors. The sensitivity of the cost functions to the parameters is explored in regions of parameter space corresponding to model small-order systems with inexpensive function evaluations and also a coupled system of nineteen equations with very expensive function evaluations. The algorithm was implemented on the Intel Hypercube, and some experimental results for the model problems with stripwise decompositions are presented and compared with the theory. In the context of computational combustion problems, multiprocessors of either message-passing or shared-memory type may be employed with stripwise decompositions to realize speedup of O(n), where n is mesh resolution in one direction, for reasonable n

Theoretical studies in support of the 3M-vapor transport (PVTOS-) experiments

Results are reported for a preliminary theoretical study of the coupled mass-, momentum-, and heat-transfer conditions expected within small ampoules used to grow oriented organic solid (OS-) films, by physical vapor transport (PVT) in microgravity environments. It is show that previous studies made restrictive assumptions (e.g., smallness of delta T/T, equality of molecular diffusivities) not valid under PVTOS conditions, whereas the important phenomena of sidewall gas creep, Soret transport of the organic vapor, and large vapor phase supersaturations associated with the large prevailing temperature gradients were not previously considered. Rational estimates are made of the molecular transport properties relevant to copper-phthalocyanine monomeric vapor in a gas mixture containing H2(g) and Xe(g). Efficient numerical methods have been developed and are outlined/illustrated here to making steady axisymmetric gas flow calculations within such ampoules, allowing for realistic realistic delta T/T(sub)w-values, and even corrections to Navier-Stokes-Fourier 'closure' for the governing continuum differential equations. High priority follow-on studies are outlined based on these new results

Lecture 02: Tile Low-rank Methods and Applications (w/review)

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the smaller scales of the past because we could afford to do so. We present innovations that allow to approach lin-log complexity in storage and operation count in many important algorithmic kernels and thus create an opportunity for full applications with optimal scalability