Abstract
The study resulting in this paper applied a parallel algorithm based on a fourth-order compact scheme and suitable for parallel implementation of scientific/engineering systems. The particular system used for demonstration in the study was a time-dependendent system solved in parallel on a 2-head-node, 224-compute-node Apple Xserve G5 multiprocessor. The use of the approximation scheme, which necessitated discretizing in both space and time with h x space width and h t time step, produced a linear tridiagonal, almost-Toeplitz system. The solution used p processors with p ranging from 3 to 63. The speedups, s p , approached the limiting value of p only when p was small but yieldd poor computations errors which became progressively better as p increases. The parallel solution is very accurate having good speedups and accuracies but only when p is within reasonable range of values.
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References
Krause, E., Kordulla, W.: Fourth-Order Mehrstellen Integration for Three-Dimensional Turbulent Boundary Layers. In: AIAA Comp. Fluid Dyn. Conf. (1973)
Gustaffsson, B.: Time Compact Difference Methods for Wave Propagation in Discontinuous Media, Tech. Reports., Dept. of Inf. Tech., Uppsala Univ., 2003-023 (2003)
Cohen, G., Joly, P.: Construction and Analysis of Fourth-Order Finite Difference Schemes for the Acoustic Wave Equation in Nonhomogeneoues Media. SIAM J. Num. Anal. 33 (1996)
Hirsh, R.: Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by a Compact Differencing Technique. J. Comp. Phys. 19, 1 (1975)
Tolstykh, A.I.: Higher Order Accurate Non-Centered Compact Difference Schemes for Fluid Dynamics Applications. Series on Adv. in Math. for Appl. Sc. 21 (1994)
Ratnesh, K., Shukla, S.: Derivation of High-Order Compact Finite Difference Schemes for Non-Uniform Grid Using Polynomial Interpolation. Jour. Comp. Phys. 204, 2 (2005)
Joslin, R., Streett, C., Chang, C.: Validation of Three-Dimensional Incompressible Spatial Direct Numerical Simulation Code, NASA Tech. Report, TP-3025, NASA Langley Research Center (1992)
Spotz, W., Garey, G.: High-Order Compact Scheme for the Stream-function Vorticity Equations. Int’l J. for Num. Method. in Eng. 38, 20 (1995)
Haiwei, S., Kang, N., Zhang, J., Carlson, E.: A Fourth-Order Comapct Difference Scheme on Face Centered Cubic Grids with Multigrid Method for Solving 2D Convection Diffusion Equation. Math. and Comp. in Simul. 63, 6 (2003)
Lambiotte, J., Voigt, R.: The Solution of Tridiagonal Linear Systems on CDC Star-100 Computer. ACM Trans. Math. Soft. 10 (1984)
Ortega, J., Voigt, R.: Solution of Partial Differential Equations on Vector and Parallel Computers. SIAM Review (1985)
Spotz, W.: Accuracy and Performance of Numerical Wall-Boundary Conditions for Steady 2D Incompressible Stream-Function Vorticity. Int’l J. for Num. Meth. in Fluids 28, 4 (1998)
Berikelashvili, G., Gupta, M.M., Mirianashvili, M.: Convergence of Fourth Order Compact Difference Schemes for Three-Dimensional Convection Diffusion Equations. SINUM (SIAM Jour. on Num. Anal.)Â 45, 1 (2007)
Spotz, W., Garey, G.: Formulation and Experiments with High-Order Compact Schemes for Nonuniform Grids. Int’l J. for Heat & Fluid Flow 8, 3 (1998)
Gustafsson, B.: The Convergence Rate for Difference Approximations to Mixed Initial Boundary Value Problems. Math. Comp. 29, 396 (1975)
Gustafsson, B.: The Convergence Rate for Difference Approximations to General Mixed Initial Boundary Value Problems. SIAM J. Numer. Anal. 18, 179 (1981)
Abarbanel, S., Chertock, A.: Strict Stability of High-Order Compact Implicit Finite Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs. J. of Comp. Phys. 160 (2000)
Kreiss, O., Oliger, J.: Methods for the Approximate Solution of Time-Dependent Problems, GARP Report, No. 10 (1973)
Zhang, J.: Multigrid Solution of the Convection-Diffusion Equation with High Reynolds Number. In: Proc. Copper Mountain Conf. on Iter. Meth. (1996)
Zhang, J.: On Convergence and performance of iterative Methods with Fourth-Order Compact Schemes. Num. Meth. for Partial Diff. Eqns. 14 (1998)
Orszak, S., Isreal, M.: Numerical Simulation of Viscous Incompressible Flows. Annual Rev. of Fluid Mech. 6 (1974)
Hirsh, R.: Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by a Compact Differencing Technique. J. Comp. Phys. 19, 1 (1975)
Rubin, S., Khosla, P.: High-Order Numerical Solutions Using Cubic Splines, NASA CR-2653 (1976)
Rubin, S., Khosla, P.: High-Order Numerical Methods Derived from Three-Point Polynomial Interpolation, NASA CR-2735 (1976)
Adam, Y.: A Hermitian Finite Difference Method for the Solution of Parabolic Equations. Comp & Math. Appl. 1 (1975)
Adam, Y.: Highly Accurate Compact Implicit Mehods and Boundary Conditions. J. Comp. Phys. 24 (1977)
Gupta, M., Manohar, R., Stephenson, J.: A Single Cell High Order Scheme for the Convection-Diffusion Equation with Variable Coefficients. Int’l J. Num. Meth. Fluids 4 (1984)
Gupta, M.: High Accuracy Solutions of Incompressible Navier-Stokes Equations. J. Comp. Phys. 93 (1991)
Yavneh, I.: Analysis of Fourth-Order Compact Scheme for Convection-Diffusion. J. Comp. Phys. 133 (1997)
Weinan, E., Liu, J.: Essentially Compact Schemes for Unsteady Viscous Incompressible Flows. J. Comp. Phys. 126 (1996)
Stone, H.: An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations. J. of ACMÂ 20 (1973)
Stone, H.: Parallel Tridiagonal Solvers. Assoc. Comp. Mach. Trans. Soft 1 (1975)
Hockney, R.: A Fast Direct Solution of Poisson’s Equation using Fourier Analysis. J. of ACM 12 (1965)
Hockney, R., Jeshoppe, C.R.: Parallel Computers 2: Architecture, Programming, and Algorithm, 2nd edn. Inst. of Physics Pub., Bristol, Philadelphia (1988)
Sun, X., Joslin, R.: A Parallel Algorithm for Almost-Toeplitz Tridiagonal Systems. Int’l Jour. of High Speed Comp. 4 (1995)
Sun, X., Gustafson, J.: Toward a Better Parallel Performance Metric. Par. Comp. 17 (1991)
Laurie, D., Sameh, A.: The Computation and Communication Complexity of a Parallel Banded System Solver. ACM Trans. Math. Soft. 10 (1984)
Wang, H.: A Parallel Method for Tridiagonal Equations. ACM Trans. Math. Soft. 7 (1981)
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Akpan, O.H. (2007). On a High-Order Compact Scheme and Its Utilization in Parallel Solution of a Time-Dependent System on a Distributed Memory Processor. In: Li, K., Jesshope, C., Jin, H., Gaudiot, JL. (eds) Network and Parallel Computing. NPC 2007. Lecture Notes in Computer Science, vol 4672. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74784-0_1
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