Multi-Model Multi-Domain Computational Methods: Distinctives

Computational aerodynamics and computational acoustics are among the most well developed of all of the computational sciences, due to the economic importance of reliable modeling in these areas and due to their attractive intellectual legacies. What can a small NSF Project add to on-going efforts in these areas funded at billions of dollars annually in industrial and government laboratories around the world?

multidisciplinarity. Within a small group of nine co-principal investigators, we are (by training) 4 mathematicians, 3 computer scientists, and 2 engineers; and we are by sector 5 professors, 3 laboratory scientists, and 1 industrial scientist. The comcommitant diversity of perspective on which abstractions are worth supporting (theoretically or in software) and which ones are easily supportable is enriching. The mathematicians and computer scientists have been led in interesting directions by the engineers, the engineers in turn have been thrilled by the enhanced problem solving capacities generated during the project. Without the (initially forced) interactions between the disciplines, our methodological results would be less interesting and our computational analysis capabilities would be less powerful.

algorithmic progressivity. We are firmly committed to providing robust parallel implicit solution capabilities to fluid mechanical problems ---

robust because we seek to establish our methods in the applications community;

parallel because of the need to render sufficiently routine for design purposes the solution of truly large analysis problems;

implicit because of the inevitability of multiple scales, which render explicit methods ineffective after some scale ratio.

Starting with the imperative of parallelism, we consider only preconditioners that possess concurrency at least to a degree equal to the number of processors and we carefully find and address impediments to scalability. Our pseudo-transient Newton-Krylov-Schwarz methods face only multigrid schemes of FAS type as rivals in effectiveness in the domain of implicit methods; and with respect to parallelizability and robustness, we regard FAS multigrid not ready for quotidian engineering use. There is not yet a general purpose parallel FAS multigrid library because there is not yet a general purpose algorithm. Pseudo-transient NKS can retreat to small timesteps, if necessary, to provide nonlinear and linear robustness. Pseudo-transient NKS can also employ multigrid as a subdomain preconditioner. This is, in fact, recommended when available and when the subdomains problems are large.

software progressivity. We are firmly committed to providing extensible and portable software ---

extensible because improved methods and combinations of methods, some of them problem-specific, are always being developed (and probably always will be);

portable because quality software must outlive the machines on which it executes in order to amortize its development cost and because users in search of high-end resources must be flexible in how those needs are met.

We employ object-oriented practices in order to provide extensibility with maximal reuse of and minimal disruption to established code. Recognizing that for large problems it is the data transfers within the horizontally (network) and vertically (cache) extended memory system, and not the processor power that determines performance, we refine the software in an ongoing way, on the basis of performance profiling, in order to exploit all known devices for per-processor and inter-processor efficiency. Since the optimal algorithmic tuning is architecture-dependent and problem-dependent, we supply rich run-time options for tuning by specialists, with conservative defaults for nonspecialists. In the spirit of matrix-free NKS methods, we provide for simultanteous multiple representations of the discrete operators required in a simulation --- both mathematically and in data structure implementation. In an effort to induce applications users to abandon the solver aspects of their legacy codes (which are usually nonscalable) while "rescuing" the highest value discretization aspect (which is nearly always scalable), we are continually adding features to make our software PDE-oriented, such as:

"gridfunction" objects more general than mathematical vectors, together with functions on the gridfunctions, such as transformation between formulations and graphical interrogation;

mesh sequencing operations;

time-step control options.

affiliations. Our contacts within the federal agencies and industrial communities that are the major consumers of high-performance scientific computing and our contacts within the international scientific and engineering communities give us the platform to transfer our technologies. We have roles in important federal programs such as ASCI and HSCT, in engineering and mathematical professional societies like AIAA and SIAM, and in international conferences that dominate in the advancement of our central interests such as the International Conferences on Domain Decomposition and Supercomputing'9X. Our university base gives us access to creative graduate students, which we are training in an interdisciplinary spirit. Inter-institutional ties have been strengthened by visiting or full-time employment relationships among graduate students and post-doctorals in our group.