Abstract
A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Padé approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.
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References
Aitken, A.C.: On Bernoulli's numerical solution of algebraic equations. Proc. Royal Soc. Edinburgh 46 (1926) 289–305.
Henrici, P.: Elements of Numerical Analysis. John Wiley & Sons, Inc., New York, 1964.
Shanks, D.: Nonlinear transformations of divergent and slowly convergent sequences. J. Math. and Phys. 34 (1955) 1–42.
Johnson, R. C.: Alternative approach to Padé approximants. Padé Approximants and Their Applications (ed. by P. R. Graves-Morris). Academic Press, London, 1973; 53–67.
Genz, A. C.: Applications of the ε-algorithm to quadrature problems. Padé Approximants and Their Applications (ed. P. R. Graves-Morris). Academic Press, London, 1973; 105–115.
Wynn, P.: On a device for computing the em(Sn) transformation. Mathematical Tables and Other Aids to Computation 10 (1956) 91–96.
Wynn, P.: Acceleration techniques for iterated vector and matrix problems. Mathematics of Computation 16 (1962) 301–322.
Martin, E. D.; Lomax, H.: Rapid finite-difference computation of subsonic and slightly supercritical aerodynamic flows. Presented as a portion of AIAA Paper No. 74-11, 1974. Also AIAA Jour. 13 (1975) 579–586.
Martin, E. D.: Progress in application of direct elliptic solvers to transonic flow computations. Aerodynamic Analyses Requiring Advanced Computers, Part II, NASA SP-347, 1975; 839–870.
Martin, E. D.: A fast semidirect method for computing transonic aerodynamic flows. AIAA 2nd Computational Fluid Dynamics Conference Proceedings, 1975; 162–174.
Bellman, R.: Perturbation Techniques in Mathematics, Physics, and Engineering. Holt, Rinehart and Winston, New York, 1964.
Buneman, O.: A compact non-iterative Poisson solver. SUIPR Rept. 294, Inst. for Plasma Research, Stanford Univ., Stanford, Calif., 1969.
Hockney, R. W.: The potential calculation and some applications. Methods in Computational Physics, Vol. 9 (ed. by B. Alder, S. Fernbach, and M. Rotenburg). Academic Press, New York, 1970; 135–211.
Buzbee, B. L.; Golub, G. H.; Nielson, C. W.: On direct methods for solving Poisson's Equations. SIAM J. Numer. Anal. 7 (1970) 627–656.
Lomax, H.; Martin, E. D.: Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations. J. Comp. Phys. 15 (1974) 55–80.
Hafez, M. M.; Cheng, H. K.: Convergence acceleration and shock fitting for transonic aerodynamics computations. AIAA Paper No. 75-51, 1975.
Concus, P.; Golub, G. H.: Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAM J. Numer. Anal. 10 (1973) 1103–1120.
Lomax, H.; Martin, E. D.: Variants and extensions of a fast direct numerical Cauchy-Riemann solver, with illustrative applications. NASA TN D-7934, 1975.
Murman, E. M.; Cole, J. D.: Calculation of plane transonic flows. AIAA Jour. 9 (1971) 114–121.
Murman, E. M.; Krupp, J. A.: Solution of the transonic potential equation using a mixed finite difference system. Lecture Notes in Physics 8 (ed. by M. Holt). Springer-Verlag, Berlin, 1971; 199–205.
Murman, E. M.: Analysis of embedded shock waves calculated by relaxation methods. AIAA Jour. 12 (1974) 626–633.
Murman, E. M.; Bailey, F. R.; Johnson, M. H.: TSFOIL-A computer code for 2-D transonic calculations, including wind-tunnel wall effects and wave-drag evaluation. Aerodynamic Analyses Requiring Advanced Computers, Part II, NASA SP-347, 1975; 769–788.
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Dale Martin, E. (1976). A technique for accelerating iterative convergence in numerical integration, with application in transonic aerodynamics. In: Cabannes, H. (eds) Padé Approximants Method and Its Applications to Mechanics. Lecture Notes in Physics, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015663
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DOI: https://doi.org/10.1007/BFb0015663
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