High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

Journal of Computational Physics

Volumn 229, Issue 6, 2010, Pages 2009-2033

  Bruno, Oscar P ^a   Lyon, Mark ^b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF NEW HAMPSHIRE   (United States)

Author keywords

ADI; Complex geometry; Fourier continuation; Fourier series; Numerical method; Partial Differential Equation; Spectral method; Unconditional stability

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; STABILITY;

ALTERNATING DIRECTION IMPLICIT; COMPLEX GEOMETRIES; FOURIER; FOURIER CONTINUATION; HIGH-ORDER; HIGHER-ORDER; SMOOTH DOMAINS; SPECTRAL METHODS; UNCONDITIONAL STABILITY; UNCONDITIONALLY STABLE;

FOURIER SERIES;

EID: 73649095076     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.11.020     Document Type: Article

References (69)

1. Peaceman D., and Rachford Jr. H. The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3 1 (1955) 28-41
2. Douglas Jr. J. Alternating direction methods for three space variables. Numer. Math. 4 (1962) 41-63
3. Douglas Jr. J., and Pearcy C. On convergence of alternating direction procedures in the presence of singular operators. Numer. Math. 5 (1963) 175-184
4. Douglas Jr. J., and Gunn J. A general formulation of alternating direction methods. Part I. Parabolic and hyperbolic problems. Numer. Math. 6 (1964) 428-453
5. Douglas Jr. J., and Rachford Jr. H. On the numerical solution of heat conduction problems in two and three space variables. T. Am. Math. Soc. 82 2 (1956) 421-439
6. O. Bruno, Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics, in: M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne (Eds.), Topics in Computational Wave Propagation Direct and Inverse Problems Series, Lecture Notes in Computational Science and Engineering, vol. 31, 2003, pp. 43-82.
7. Bruno O., Han Y., and Pohlman M. Accurate, high-order representation of complex three-dimensional surfaces via Fourier-continuation analysis. J. Comput. Phys. 227 (2007) 1094-1125
8. Boyd J. A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. J. Comput. Phys. 178 (2002) 118-160
9. Morton K., and Mayers D. Numerical Solution of Partial Differential Equations. second ed. (2005), Cambridge University Press, Cambridge
10. M. Lyon, O. Bruno, High-order unconditionally-stable FC-AD for general two-dimensional smooth domains II: spatial accuracy, stability, and the Wave Equation, submitted for publication.
11. Strikwerda J. Finite Difference Schemes and Partial Differential Equations. second ed. (2004), SIAM, Philadelphia
12. Namiki T. A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microw. Theory 47 10 (1999) 2003-2007
13. Lees M. Alternating direction methods for hyperbolic differential equations. J. Soc. Ind. Appl. Math. 10 4 (1962) 610-616
14. Mohanty R., and Jain M. An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation. Numer. Meth. Part. Differ. Equat. 17 (2001) 684-688
15. Zheng F., Chen Z., and Zhang J. A finite-difference time-domain method without the Courant stability conditions. IEEE Microw. Guided Wave Lett. 9 11 (1999) 441-443
16. Averbuch A., and Vozovoi L. Two-dimensional parallel solver for the solution of Navier-Stokes equations with constant and variable coefficients using ADI on cells. Parallel Comput. 24 (1998) 673-699
17. Badcock K., and Richards B. Implicit time-stepping methods for the Navier-Stokes equations. AIAA J. 34 3 (1996) 555-559
18. Widlund O. On the effects of scaling of the Peaceman-Rachford method. Math. Comput. 25 113 (1971) 33-41
19. Widlund O. On the rate of convergence of an alternating direction implicit method in a noncommutative case. Math. Comput. 20 96 (1966) 500-515
20. Hundsdorfer W., and Verwer J.G. Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Math. Comput. 53 187 (1989) 81-101
21. Gunn J. On the two-stage iterative method of Douglas for mildly nonlinear elliptic difference equations. Numer. Math. 6 (1964) 243-249
22. Fornberg B., Zuev J., and Lee J. Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equations. J. Comput. Appl. Math. 200 (2007) 178-192
23. Fairweather G., and Mitchell A. A high accuracy alternating direction method for the wave equation. J. Inst. Math. Appl. 1 (1965) 309-316
24. Mitchell A., and Fairweather G. Improved forms of the alternatin direction methods of Douglas, Peaceman and Rachford for solving parabolic and elliptic equations. Numer. Math. 6 (1964) 285-292
25. Tian Z., and Ge Y. A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems. J. Comput. Appl. Math. 198 (2007) 268-286
26. You D. A high-order Padé ADI method for unsteady convection-diffusion equations. J. Comput. Phys. 214 (2006) 1-11
27. Kantartzis N., Zygiridis T., and Tsiboukis T. An unconditionally stable higher order ADI-FDTD technique for the dispersionless analysis of generalized 3-D EMC structures. IEEE Trans. Magn. 40 2 (2004) 1436-1439
28. Dyksen W. Tensor product generalized ADI methods for separable elliptic problems. SIAM J. Numer. Anal. 24 1 (1987) 59-75
29. Lynch R., Rice J., and Thomas D. Direct solution of partial difference equations by tensor product methods. Numer. Math. 6 (1964) 185-199
30. Carpenter M., Gottlieb D., and Saul A. The stability of numerical boundary treatments for compact high-order finite-difference schemes. J. Comput. Phys. 108 (1993) 272-295
31. Abarbanel S., and Chertock A. Strict stability of high-order compact implicit finite-difference schemes: the role of boundary conditions for hyperbolic PDEs, I. J. Comput. Phys. 160 (2000) 42-66
32. Abarbanel S., Chertock A., and Yefet A. Strict stability of high-order compact implicit finite-difference schemes: the role of boundary conditions for hyperbolic PDEs, II. J. Comput. Phys. 160 (2000) 67-87
33. Karaa S., and Zhang J. High order ADI method for solving unsteady convection-diffusion problems. J. Comput. Phys. 198 (2004) 1-9
34. Clavero C., Gracia J., and Jorge J. A uniformly convergent alternating direction hodie finite difference scheme for 2d time-dependent convection-diffusion problems. IMA.J. Numer. Anal. 26 (2006) 155-172
35. Alpert B., Greengard L., and Hagstrom T. An integral evolution formula for the wave equation. J. Comput. Phys. 162 2 (2000) 536-543
36. Abarbanel S., Ditkowski A., and Yefet A. Bounded error schemes for the wave equation on complex domains. J. Sci. Comput. 26 (2006) 67-81
37. Abarbanel S., and Ditkowski A. Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes. J. Comput. Phys. 133 (1997) 279-288
38. Elghaoui M., and Pasquetti R. A spectral embedding method applied to the advection-diffusion equation. J. Comput. Phys. 125 (1996) 464-476
39. Lombard B., Piraux J., Gelis C., and Virieux J. Free and smooth boundaries in 2-d finite-difference schemes for transient elastic waves. Geophys. J. Int. 172 (2008) 252-261
40. Griffith B., and Peskin C. On the order of accuracy of the immersed boundary method: higher order convergence rates for suffciently smooth problems. J. Comput. Phys. 208 (2005) 75-105
41. Lai M., and Peskin C. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys. 160 (2000) 705-719
42. Peskin C. The immersed boundary method. Acta Numer. 11 (2002) 479-517
43. Johansen H., and Colella P. A cartesian grid embedded boundary method for Poisson's equation on irregular domains. J. Comput. Phys. 147 (1998) 60-85
44. Kreiss H., and Petersson N. A second order accurate embedded boundary method for the wave equation with dirichlet data. SIAM J. Sci. Comput. 27 (2006) 1141-1167
45. Ditkowski A., Dridi K., and Hesthaven J. Convergent Cartesian grid, methods for maxwell's equations in complex geometries. J. Comput. Phys. 170 (2001) 39-80
46. Li J., and Greengard L. High order marching schemes for the wave equation in complex geometry. J. Comput. Phys. 198 (2004) 295-309
47. Sabetghadam F., Sharafatmandjoor S., and Norouzi F. Fourier spectral embedded boundary solution of the Poisson's and Laplace equations with dirichlet boundary conditions. J. Comput. Phys. 228 (2009) 55-74
48. Eckhoff K. On a high order numerical method for solving Partial Differential Equations in complex geometries. J. Sci. Comput. 12 2 (1997) 119-138
49. Næss O., and Eckhoff K. A modified Fourier Galerkin method for the Poisson and Helmholtz equations. J. Sci. Comput. 17 (2002) 529-539
50. Zhao G., and Liu Q. The unconditionally stable pseudospectral time-domain (PSTD) method. IEEE Microw. Wireless Comp. Lett. 13 11 (2003) 475-477
51. Bialecki B., and Fernandes R. An alternating-direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons. SIAM J. Sci. Comput. 28 3 (2006) 1054-1077
52. Cooper K., and Prenter P. Alternating direction for separable elliptic Partial Differential Equations. SIAM J. Numer. Anal. 28 3 (1991) 711-727
53. Cooper K., McArthur K., and Prenter P. Alternating direction collocation for irregular regions. Numer. Meth. Part. Differ. Equat. 12 (1996) 147-159
54. Canuto C., Hussaini M., Quarteroni A., and Zang T. Spectral Methods Fundamentals in Single Domains, Scientific Computation (2006), Springer, Berlin
55. Canuto C., Hussaini M., Quarteroni A., and Zang T. Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation (2007), Springer, Berlin
56. Gottlieb D., and Shu C.-W. On the Gibbs phenomenon and its resolution. SIAM Rev. 39 4 (1997) 644-668
57. Gelb A., and Tanner J. Robust reprojection methods for the resolution of the Gibbs phenomenon. Appl. Comput. Harmon. Anal. 20 (2006) 3-25
58. Boyd J.P., and Ong J.R. Exponentially-convergent strategies for defeating the Runge Phenomenon for the approximation of non-periodic functions, part I: single-interval schemes. Commun. Comput. Phys. 5 2-4 (2009) 484-497
59. Frigo M., and Johnson S. The design and implementation of FFTW3. Proc. IEEE 93 2 (2005) 216-231
60. Å. Björck, Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1996.
61. Barnard R., Dahlquist G., Pearce K., Reichel L., and Richards K. Gram polynomials and the kummer function. J. Approx. Theory 94 (1998) 128-143
62. Demmel J. Applied Numerical Linear Algebra (1997), SIAM, Philadelphia
63. M. Lyon, High-order unconditionally-stable FC-AD PDE solvers for general domains, Ph.D. thesis, California Institute of Technology, 2009.
64. J. Weideman, S. Reddy, A MATLAB differentiation matrix suite, ACM T. Math. Software 26(4) (2000) 465-519. url http://doi.acm.org/10.1145/365723.365727.
65. StoerJ B.R. Introduction to Numerical Analysis. second ed. (2004), Springer, New York
66. Wachspress E. Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics (1971), Prentice-Hall, Englewood Cliffs, NJ
67. Young D. Iterative Solution of Large Linear Systems (1971), Academic Press, New York
68. Quarteroni A., Riccardo S., and Saleri F. Numerical Mathematics, Texts in Applied Mathematics (2000), Springer, New York
69. Benzi M. Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182 (2002) 418-477