A grid-free abstraction of the Navier-Stokes equations in Fortran 95/2003 | ACM Transactions on Mathematical Software
- ️KassinosStavros C.
- ️Sat Jan 19 2008
Article No.: 2, Pages 1 - 33
Abstract
Computational complexity theory inspires a grid-free abstraction of the Navier-Stokes equations in Fortran 95/2003. A novel complexity analysis estimates that structured programming time grows at least quadratically with the number of program lines. Further analysis demonstrates how an object-oriented strategy focused on mathematical objects renders the quadratic estimate scale-invariant, so the time required for the limiting factor in program development (debugging) no longer grows as the code grows. Compared to the coordinate-free C++ programming of Grant et al. [2000], grid-free Fortran programming eliminates a layer of procedure calls, eliminates a related need for the C++ template construct, and offers a shorter migration path for Fortran programmers. The grid-free strategy is demonstrated by constructing a physical-space driver for a Fourier-space Navier-Stokes solver. Separating the expression of the continuous mathematical model from the discrete numerics clarifies issues that are otherwise easily conflated. A run-time profile suggests that grid-free design substantially reduces the fraction of the procedures that significantly impact runtime, freeing more code to be structured in ways that reduce development time. Applying Amdahl's law to the total solution time (development time plus run time) leads to a strategy that negligibly impacts development time but achieves 58% of the maximum possible speedup.
References
[1]
Akin, E. 2003. Object-Oriented Programming via Fortran 90/95. Cambridge University Press, Cambridge, UK.
[2]
Barton, J. J. and Nackman, L. R. 1994. Scientific and Engineering C++: An Introduction with Advanced Examples. Addison-Wesley. Boston, MA.
[3]
Bernholdt, D. E., Allan, B. A., Armstrong, R., Bertrand, F., Chiu, K., Dahlgren, T. L., Damevski, K., Elwasif, W. R., Epperly, T. G. W., Govindaraju, M., Katz, D. S., Kohl, J. A., Krishnan, M., Kumfert, G., Larson, J. W., Lefantzi, S., Lewis, M. J., Malony, A. D., McInnes, L. C., Nieplocha, J., Norris, B., Parker, S. G., Ray, J., Shende, S., Windus, T. L., and Zhou, S. 2006. A component architecture for high-performance scientific computing. Int. J. High Perform. Comput. Appl. 20, 163--202.
[4]
Berryhill, J. R. 2001. C++ Scientific Programming: Computational Recipes at a Higher Level. Wiley-Interscience, Hoboken, NJ.
[5]
Bronson, G. J. 2006. C++ for Engineers and Scientists 2nd Ed. Thomson Learning, Stamford, CT.
[6]
Buzzi-Ferraris, G. 1998. Scientific C++. Addison-Wesley, Boston, MA.
[7]
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. 1988. Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin, Germany.
[8]
Collins, J. B. 2004. Standardizing an ontology of physics for modeling and simulation. Proceedings of the Fall Simulation Interoperability Workshop. Orlando, FL.
[9]
Decyk, V. K., Norton, C. D., and Szymanski, B. K. 1997. Expressing object-oriented concepts in Fortran 90. ACM Fortran For. 15, 13--18.
[10]
Decyk, V. K., Norton, C. D., and Szymanski, B. K. 1997. How to express C++ concepts in Fortran 90. Scient. Program. 6, 363--390.
[11]
Decyk, V. K., Norton, C. D., and Szymanski, B. K. 1997. High performance object-oriented programming in Fortran 90. In Proceedings of the 8th SIAM Conference On Parallel Processing for Scientific Computing. M. Heath et al., Eds. Minneapolis, MN.
[12]
Decyk, V. K., Norton, C. D., and Szymanski, B. K. 1998. How to support inheritance and run-time polymorphism in Fortran 90. Comput. Phys. Comm. 115, 9--17.
[13]
Dinesh, T., Haveraaen, M., and Heering, J. 2000. An algebraic programming style for numerical software and its optimization. Scient. Program. 8, 247--259.
[14]
Fenton, N. E. and Ohlsson, N. 2000. Quantitative analysis of faults and failures in a complex software system. IEEE Trans. Softw. Eng. 26, 797--814.
[15]
Grant, P. W. Haveraaen, M. and Webster, M. F. 2000. Coordinate free programming of computational fluid dynamics problems. Scient. Program. 8, 211--230.
[16]
Hatton, L. 1997. The T experiments: Errors in scientific software. IEEE Computat. Scien. Eng. 4, 27--38.
[17]
Haveraaen, M. Madsen, V., and Munthe-Kaas, M. 1992. Algebraic programming technology for partial differential equations. In Proceedings Norsk Informatikk Konferanse, A. Maus et al., Eds. 55--68.
[18]
Kim, J., Moin, P., and Moser, R.D. 1987. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mechan. 177, 133--166.
[19]
Lefantzi, S. Ray, J., Kennedy, C. A., and Najm, H. N. 2005. A component-based toolkit for simulating reacting flows with high order spatial discretizations on structured adaptively refined meshes. Prog. Computat. Fluid Dynam. 5, 298--315.
[20]
Machiels, L. and Deville, M. O. 1997. Fortran 90: An entry into object-oriented programming for the solution of partial differential equations. ACM Trans. Math. Softw. 23, 32--49.
[21]
Markus, A. 2003. Avoiding memory leaks with derived types. ACM Fortran For. 22, 1--6.
[22]
Malawski, M., Dawid, K., and Vaidy, Sunderam. 2005. MOCCA---Towards a distributed CCA framework for metacomputing. Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium (IPDPS)---IEEE Computer Society.
[23]
McInnes, L. C., Allan, B. A., Armstrong, R., Benson, S. J., Bernholdt, D. E., Dahlgren, T. L., Diachin, L. F., Krishnan, M., Kohl, J. A., Larson, J. W., Lefantzi, S., Nieplocha, J., Norris, B., Parker, S. G., Ray, J., and Zhou, S. 2005. Parallel PDE-based simulations using the Common Component Architecture. In Numerical Solution of Partial Differential Equations on Parallel computers, A.M., Bruaset, and A. Tveito, Eds. Springer, Berlin, Germany.
[24]
Metcalf, M., Reid, J., and Cohen, M. 2004. Fortran 95/2003 Explained. Oxford University Press, Oxford, UK.
[25]
Mittal, R. and Iaccarino, G. 2005. Immersed boundary methods. Ann. Rev. Fluid Mechan. 37, 239--261.
[26]
Moin, P. 2001. Fundamentals of Engineering Numerical Analysis. Cambridge University Press, Cambridge, UK.
[27]
Oliveira, S. and Stewart, D. 2006. Writing Scientific Software: A Guide to Good Style. Cambridge University Press, Cambridge, UK.
[28]
Orszag, S. A. and Patterson Jr., G. S. 1972. Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76--79.
[29]
Rouson, D. W. I., Kassinos, S. C., Sarris, I., and Toschi, F. 2006b. Particle dispersion in magnetohydrodynamic turbulence at low magnetic Reynolds number. In Proceedings of the CTR Summer Program. Center for Turbulence Research, Stanford University, Stanford, CA.
[30]
Rouson, D. W. I., Morris, K., and Xu, X. 2005. Dynamic memory de-allocation in Fortran 95/2003 derived type calculus.Scien. Program. 13, 189--203.
[31]
Rouson, D. W. I. and Xiong, Y. 2004. Design metrics in quantum turbulence simulations: How physics influences software architecture. Scien. Program. 12, 185--196.
[32]
Rouson, D. W. I., Xu, X., and Morris, K. 2006a. Formal constraints on memory management in composite overloaded operations, Scien. Program. 14, 27--40.
[33]
Shalloway, A. and Trott, J. R. 2002. Design Patterns Explained: A New Perspective on Object-Oriented Design. Addison-Wesley.
[34]
Spalart, P. R., Moser, R. D., and Rogers, M. M. 1991. Spectral methods for the Navier-Stokes equations in one infinite and two periodic directions. J. Comput. Phys. 96, 297--324.
[35]
Stewart, G. W. 2003. Memory leaks in derived types revisited. ACM Fortran For. 22, (Dec), 25--27.
[36]
Yang, D. 2000. C++ and Object-Oriented Numeric Computing for Scientists and Engineers. Springer, Berlin, Germany.
[37]
Zhang, K., Damevski, K., Venkatachalapathy, V., and Parker, S.G. 2004. SCIRun2: A CCA framework for high performance computing. In 9th International Workshop on High-Level Parallel Programming Models and Supportive Environments (HIPS'04). Rasmussen, C., Workshop Chair, Santa Fe, NM, 72--79.
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ACM Transactions on Mathematical Software Volume 34, Issue 1
January 2008
131 pages
Copyright © 2008 ACM.
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Publication History
Published: 19 January 2008
Accepted: 01 March 2007
Revised: 01 December 2006
Received: 01 March 2006
Published in TOMS Volume 34, Issue 1
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