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barcode for .NET Spectral Methods in Software Generator PDF-417 2d barcode in Software Spectral Methods

20.7 Spectral Methods generate, create none none in none projectspdf-417 generating asp.net one of the method none none s for nding differentiation matrices, we get p p 2 32 3 11 1 4C2 2 2 4 2 2 y0 2 2 p p p p 76 7 6 1 1 p2 1 1 1 2 2 2 1 2 2 7 6 y1 7 6 2 2 2 6 76 7 p p .1/ 1 1 7 6 y2 7 D y i D 6 2 0 2 6 2 2 p p p p p 76 7 6 76 7 1 1 1 1 4 1C 2 2 2 2 2 1 C 2 25 4 y 3 5 2 2 p p 1 11 4C2 2 2 4 2 2 y4 2 2 (20.7.

50) and 17 p 6 65 C 3 2 6 1 D .2/ y i D 6 6 p 6 45 3 2 5 2 20 p 6 2 14 4 2 20 C 6 2 p 18 6 6 6 18 20 p 20 C 6 2 2 4 14 p 6 2 32 3 y0 p 76 7 5 3 2 7 6y1 7 76 7 1 7 6y2 7 p 76 7 76 7 5 C 3 25 4 y 3 5 5 17 y4 (20.7.

51) Requiring that the differential equation hold at the interior collocation points xk , k D 1; 2; 3, uses the middle three rows of these matrices. Enforcing the boundary conditions y0 D y4 D 0 means we don t need the rst and last columns. So equation (20.

7.45) gives p p 32 3 2 p 3 2 2 16 C 1 2 6 C 2 2 1 2 y1 2 2 p 76 7 6 p 7 6 (20.7.

52) 2 8 4 C 2 5 4 y 2 5 D 4 25 4 4 p p p 1 1 2C 2 2 6 2 2 16 2 2 y3 with solution 2 y1 y3 3 2 101. Code 128 Code Set B 350 101 350. 6 7 6 4 y2 5 D 4 3 13 2 350 7 13 5 25 p 13 2 350. (20.7.53).

The exact solutio n (20.7.47) gives for example y.

x D 0/ D 0:52065, compared with y2 D 0:52000. Not bad for ve grid points! The real point, however, is that the error is about 10 16 for N D 16. With a second-order nite difference scheme, the error would go down by only a factor of 10 or so with this increase in N .

. 20.7.13 Multidimensional Spectral Methods For a time-depend ent problem, the simplest approach is the method of lines. Expand the solution as X Cj .x/yj .

t / (20.7.54) y.

t; x/ D. where now the coe none none f cients yj are functions of time. Then X .1/ @y @y D yi ; D P Dij yj ; @t i @x i.

etc. (20.7.55).

20. Partial Differential Equations You get a system none none of ODEs in t for the yj , which you can solve in the standard way. Runge-Kutta is a good method to start with. Problems with two or three spatial dimensions are usually handled by making expansions along each dimension separately: X uij k Ci .

x/Cj .y/Ck .z/ (20.

7.56) u.x; y; z/ D.

ij k Elliptic equation none none s give simultaneous algebraic equations for the coef cients that are typically solved with iterative methods because of the large number of variables. See [11] for an example and references to the literature..

CITED REFERENCES AND FURTHER READING: Hesthaven, J., Gottlieb, S., and Gottlieb, D.

2007, Spectral Methods for Time-Dependent Problems (New York: Cambridge University Press), 9.[1] Gottlieb, D., and Orszag, S.

A. 1977, Numerical Analysis of Spectral Methods: Theory and Applications (Philadelphia: S.I.

A.M.).

[2] [A classic, and still somewhat useful.] Canuto, C., Hussaini, M.

Y., Quarteroni, A., and Zang, T.

A. 1988, Spectral Methods in Fluid Dynamics (Berlin: Springer).[3] [Standard reference for uid dynamics applications, but applicable to other areas.

] Boyd, J.P. 2001, Chebyshev and Fourier Spectral Methods, 2nd ed.

(New York: Dover Publications). Available at http://www-personal.engin.

umich.edu/~jpboyd.[4] [Best single book: complete, and not too formal.

] Fornberg, B. 1996, A Practical Guide to Pseudospectral Methods (New York: Cambridge University Press).[5] [Good for getting started, but not for large-scale problems.

] Fornberg, B. 1998, Calculation of Weights in Finite Difference Formulas, SIAM Review vol. 40, pp.

685 691.[6] Baltensperger, R., and Trummer, M.

R. 2003, Spectral Differencing with a Twist, SIAM Journal on Scienti c Computing, vol. 24, pp.

1465 1487.[7] Matsushima, T., and Marcus, P.

S. 1995, A Spectral Method for Polar Coordinates, Journal of Computational Physics vol. 120, pp.

365 374.[8] Matsushima, T., and Marcus, P.

S. 1997, A Spectral Method for Unbounded Domains, Journal of Computational Physics vol. 137, pp.

321 345.[9] Rawitscher, G.H.

1991, Accuracy Analysis of a Bessel Spectral Function Method for the Solution of Scattering Equations, Journal of Computational Physics vol. 94, pp. 81 101.

[10] Pfeiffer, H.P., Kidder, L.

E., Scheel, M.A.

, and Teukolsky, S.A. 2003, A Multidomain Spectral Method for Solving Elliptic Equations, Computer Physics Communications, vol.

152, pp. 253 273.[11] Bj rhus, M.

1995, The ODE Formulation of Hyperbolic PDEs Discretized by the Spectral Collocation Method, SIAM Journal on Scienti c Computing, vol. 16, pp. 542 557.

[Describes a good algorithm for hyperbolic equations.].
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