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Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems
A Comparison of some numerical conformal mapping methods for simply and multiply connected domains
Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA |
This paper compares some methods for computing conformal maps from simply and multiply connected domains bounded by circles to target domains bounded by smooth curves and curves with corners. We discuss the use of explicit preliminary maps, including the osculation method of Grassmann, to first conformally map the target domain to a more nearly circular domain. The Fourier series method due to Fornberg and its generalization to multiply connected domains are then applied to compute the maps to the nearly circular domains. The final map is represented as a composition of the Fourier/Laurent series with the inverted explicit preliminary maps. A method for systematically removing corners with power maps is also implemented and composed with the Fornberg maps. The use of explict maps has been suggested often in the past, but has rarely been carefully studied, especially for the multiply connected case. Using Fourier series to represent conformal maps from domains bounded by circles to more general domains has certain computational advantages, such as the use of fast methods. However, if the target domain has elongated sections or corners, the mapping problems can suffer from severe ill-conditioning or loss of accuracy. The purpose of this paper is to illustrate some of these practical possibilites and limitations.
References:
[1] |
L. Ahlfors,
Complex Analysis, third edition, McGraw-Hill, New York, 1979. |
[2] |
M. Badreddine,
Comparison of Some Numerical Conformal Mapping Methods for Simply and Multiply Connected Domains, Ph. D dissertation, Wichita State University, 2016. |
[3] |
N. Benchama, T. DeLillo, T. Hrycak and L. Wang,
A simplified Fornberg-like method for the conformal mapping of multiply connected regions-comparisons and crowding, J. Comput. Appl. Math., 209 (2007), 1-21.
doi: 10.1016/j.cam.2006.10.030. |
[4] |
T. K. DeLillo,
A Comparison of Some Numerical Conformal Mapping Methods, PhD dissertation, Courant Institute, NYU, 1985. |
[5] |
T. K. DeLillo, On the use of numerical conformal mapping methods in solving boundary value problems for the Laplace equation, in Advances in Computer Methods for Partial Differential Equations-Ⅶ, eds. R. Vichnevetsky, D. Knight, and G. Richter, Seventh IMACS Symposium Proceedings, Rutgers University, (1992), 190-194. |
[6] |
T. K. DeLillo, Comparisons of some numerical conformal mapping methods, in Proceedings of the 14th IMACS World Congress on Computation and Applied Mathematics, Vol. 1, ed. W. F. Ames, Georgia Institute of Technology, Atlanta, Georgia, (1994), 115-118. |
[7] |
T. DeLillo,
The accuracy of numerical conformal mapping methods: A survey of examples and results, SIAM J. Numer. Anal., 31 (1994), 788-812.
doi: 10.1137/0731043. |
[8] |
T. DeLillo, Tutorial on Fourier series methods for numerical conformal mapping of smooth domains, 2014, http://www.math.wichita.edu/~delillo/TD_tutorial.pdf |
[9] |
T. DeLillo and A. Elcrat,
A comparison of some numerical conformal mapping methods for exterior regions, SIAM J. Sci. Stat. Comput., 12 (1991), 399-422.
doi: 10.1137/0912022. |
[10] |
T. K. DeLillo and A. R. Elcrat,
Numerical conformal mapping methods for exterior regions with corners, J. Comput. Phys., 108 (1993), 199-208.
doi: 10.1006/jcph.1993.1175. |
[11] |
T. K. DeLillo, A. R. Elcrat and J. A. Pfaltzgraff,
Numerical conformal mapping methods based on Faber series, J. Comput. Appl. Math., 83 (1997), 205-236.
doi: 10.1016/S0377-0427(97)00099-X. |
[12] |
T. K. DeLillo and E. H. Kropf, A Fornberg-like method for the numerical conformal mapping of bounded multiply connected domains, submitted for publication. |
[13] |
T. DeLillo and J. Pfaltzgraff,
Numerical conformal mapping methods for simply and doubly connected regions, SIAM J. Sci. Comput., 19 (1998), 155-171.
doi: 10.1137/S1064827596303545. |
[14] |
T. A. Driscoll and L. N. Trefethen,
Schwarz-Christoffel Mapping, Cambridge, 2002. |
[15] |
B. Fornberg,
A numerical method for conformal mappings, SIAM J. Sci. Stat. Comput, 1 (1980), 386-400.
doi: 10.1137/0901027. |
[16] |
D. Gaier,
Konstruktive Methoden der Konformen Abbildung, Springer, Berlin, 1964. |
[17] |
E. Grassmann,
Numerical experiments with a method of successive approximation for conformal mapping, ZAMP, 30 (1979), 873-884.
doi: 10.1007/BF01590486. |
[18] |
M. H. Gutknecht,
Numerical experiments on solving Theodorsen's intergral equation for conformal maps with the fast Fourier transfomr and various nonlinear iterative methods, SIAM J. Sci. Stat. Comput., 4 (1983), 1-30.
doi: 10.1137/0904001. |
[19] |
H. Hakula, T. Quach and A. Rasila,
Conjugate function method for numerical conformal mappings, J. Comput. Appl. Math., 237 (2013), 340-353.
doi: 10.1016/j.cam.2012.06.003. |
[20] |
H. Hakula, A. Rasila and M. Vuorinen, Conformal modulus on domains with strong singularities and cusps, arXiv: 1501.06765. |
[21] |
N. D. Halsey,
Potential flow analysis of multielement airfoils using conformal mapping, AIAA J., 17 (1979), 1281-1288.
doi: 10.2514/3.61308. |
[22] |
J. Heinhold and R. Albrecht,
Zur Praxis der konformen Abbildung, Rend. Circ. Mat. Palermo Ser. 2, 3 (1954), 130-148.
doi: 10.1007/BF02849374. |
[23] |
P. Henrici,
A general theory of osculation algorithms for conformal maps, J. Linear Alg. Appl., 52/53 (1983), 361-382.
doi: 10.1016/0024-3795(83)80024-X. |
[24] |
P. Henrici,
Applied and Computational Complex Analysis, John Wiley & Sons, Inc., New York, 1988. |
[25] |
H.-P. Hoidn,
Osculation methods for the conformal mapping of doubly connected regions, ZAMP, 33 (1982), 640-652.
|
[26] |
W. D. Hoskins and P. R. King,
Periodic cubic spline interpolation using parametric splines, The Computer Journal, 15 (1972), 282-283.
|
[27] |
R. M. James,
A general class of exact airfoil solutions, AIAA J., 9 (1972), 574-580.
doi: 10.2514/3.59038. |
[28] |
W. Koppenfels and F. Stallmann,
Praxis Der Konformen Abbildung, Springer, Berlin, 1959. |
[29] |
L. Landweber and T. Miloh,
Elimination of corners in the mapping of a closed curve, J. Engrg. Math., 6 (1972), 369-375.
doi: 10.1007/BF01535197. |
[30] |
R. S. Lehman,
Development of the mapping function at an analytic corner, Pacific J. Math., 7 (1957), 1437-1449.
doi: 10.2140/pjm.1957.7.1437. |
[31] |
H. Lewy,
Developments at the confluence of anaytic boundary conditions, Univ. of California Publ. in Math., 1 (1950), 247-280.
|
[32] |
D. E. Marshall,
Conformal welding for finitely connected regions, Comput. Methods Funct. Theory, 11 (2011), 655-669.
|
[33] |
J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, Sixth edition, Jones and bartlett, W, C. Brown, 2010. http://mathfaculty.fullerton.edu/mathews/complex.html |
[34] |
D. I. Meiron, S. A. Orszag and M. Israeli,
Applications of numerical conformal mapping, J. Comput. Phys., 40 (1981), 345-360.
doi: 10.1016/0021-9991(81)90215-1. |
[35] |
R. Menikoff and C. Zemach,
Methods for numerical conformal mapping, J. Comput. Phys., 36 (1980), 366-410.
doi: 10.1016/0021-9991(80)90166-7. |
[36] |
M. Nasser,
Fast computation of the circular map, Comput. Methods Funct. Theory, 15 (2015), 187-223.
doi: 10.1007/s40315-014-0098-3. |
[37] |
M. Nasser, T. Sakajo, A. Murid and L. K. Wei,
A fast computational method for potential flows in multiply connected coastal domains, Japan J. Indust. Appl. Math., 32 (2015), 205-236.
doi: 10.1007/s13160-015-0168-6. |
[38] |
S. T. O'Donnell and V. Rokhlin,
A fast algorithm for the numerical evaluation of conformal mappings, SIAM J. Sci. Statist. Comput., 10 (1989), 475-487.
doi: 10.1137/0910031. |
[39] |
N. Papamichael and N. Stylianopoulos,
Numerical Conformal Mapping - Domain Decomposition and the Mapping of Quadrilaterals, World Scientific, Singapore, 2010. |
[40] |
R. M. Porter,
An accelerated osculation method and its application to numerical conformal mapping, Complex Variables, 48 (2003), 569-582.
doi: 10.1080/0278107031000110892. |
[41] |
W. J. Prosnak,
Computation of Fluid Motions in Multiply Connected Domains, G. Braun, Karlsruhe, 1987. |
[42] |
W. J. Prosnak,
Conformal representation of arbitrary multiconnected airfoils, Bull. Acad. Pol. Sci., 25 (1977), 25-36 (591-602).
|
[43] |
R. Wegmann,
On Fornberg's numerical method for conformal mapping, SIAM J. Numer. Anal., 23 (1986), 1199-1213.
doi: 10.1137/0723081. |
[44] |
——, Methods for numerical conformal mapping, in Handbook of Complex Analysis, Geometric Function Theory, Vol. 2, (ed. R. Kuehnau), Elsevier, Amsterdam, (2005), 351-477. |
[45] |
B. R. Williams, An exact test case for the plane potential flow about two adjacent lifting airfoils,
RAE Technical Report No. 3717, London (1973). |
show all references
References:
[1] |
L. Ahlfors,
Complex Analysis, third edition, McGraw-Hill, New York, 1979. |
[2] |
M. Badreddine,
Comparison of Some Numerical Conformal Mapping Methods for Simply and Multiply Connected Domains, Ph. D dissertation, Wichita State University, 2016. |
[3] |
N. Benchama, T. DeLillo, T. Hrycak and L. Wang,
A simplified Fornberg-like method for the conformal mapping of multiply connected regions-comparisons and crowding, J. Comput. Appl. Math., 209 (2007), 1-21.
doi: 10.1016/j.cam.2006.10.030. |
[4] |
T. K. DeLillo,
A Comparison of Some Numerical Conformal Mapping Methods, PhD dissertation, Courant Institute, NYU, 1985. |
[5] |
T. K. DeLillo, On the use of numerical conformal mapping methods in solving boundary value problems for the Laplace equation, in Advances in Computer Methods for Partial Differential Equations-Ⅶ, eds. R. Vichnevetsky, D. Knight, and G. Richter, Seventh IMACS Symposium Proceedings, Rutgers University, (1992), 190-194. |
[6] |
T. K. DeLillo, Comparisons of some numerical conformal mapping methods, in Proceedings of the 14th IMACS World Congress on Computation and Applied Mathematics, Vol. 1, ed. W. F. Ames, Georgia Institute of Technology, Atlanta, Georgia, (1994), 115-118. |
[7] |
T. DeLillo,
The accuracy of numerical conformal mapping methods: A survey of examples and results, SIAM J. Numer. Anal., 31 (1994), 788-812.
doi: 10.1137/0731043. |
[8] |
T. DeLillo, Tutorial on Fourier series methods for numerical conformal mapping of smooth domains, 2014, http://www.math.wichita.edu/~delillo/TD_tutorial.pdf |
[9] |
T. DeLillo and A. Elcrat,
A comparison of some numerical conformal mapping methods for exterior regions, SIAM J. Sci. Stat. Comput., 12 (1991), 399-422.
doi: 10.1137/0912022. |
[10] |
T. K. DeLillo and A. R. Elcrat,
Numerical conformal mapping methods for exterior regions with corners, J. Comput. Phys., 108 (1993), 199-208.
doi: 10.1006/jcph.1993.1175. |
[11] |
T. K. DeLillo, A. R. Elcrat and J. A. Pfaltzgraff,
Numerical conformal mapping methods based on Faber series, J. Comput. Appl. Math., 83 (1997), 205-236.
doi: 10.1016/S0377-0427(97)00099-X. |
[12] |
T. K. DeLillo and E. H. Kropf, A Fornberg-like method for the numerical conformal mapping of bounded multiply connected domains, submitted for publication. |
[13] |
T. DeLillo and J. Pfaltzgraff,
Numerical conformal mapping methods for simply and doubly connected regions, SIAM J. Sci. Comput., 19 (1998), 155-171.
doi: 10.1137/S1064827596303545. |
[14] |
T. A. Driscoll and L. N. Trefethen,
Schwarz-Christoffel Mapping, Cambridge, 2002. |
[15] |
B. Fornberg,
A numerical method for conformal mappings, SIAM J. Sci. Stat. Comput, 1 (1980), 386-400.
doi: 10.1137/0901027. |
[16] |
D. Gaier,
Konstruktive Methoden der Konformen Abbildung, Springer, Berlin, 1964. |
[17] |
E. Grassmann,
Numerical experiments with a method of successive approximation for conformal mapping, ZAMP, 30 (1979), 873-884.
doi: 10.1007/BF01590486. |
[18] |
M. H. Gutknecht,
Numerical experiments on solving Theodorsen's intergral equation for conformal maps with the fast Fourier transfomr and various nonlinear iterative methods, SIAM J. Sci. Stat. Comput., 4 (1983), 1-30.
doi: 10.1137/0904001. |
[19] |
H. Hakula, T. Quach and A. Rasila,
Conjugate function method for numerical conformal mappings, J. Comput. Appl. Math., 237 (2013), 340-353.
doi: 10.1016/j.cam.2012.06.003. |
[20] |
H. Hakula, A. Rasila and M. Vuorinen, Conformal modulus on domains with strong singularities and cusps, arXiv: 1501.06765. |
[21] |
N. D. Halsey,
Potential flow analysis of multielement airfoils using conformal mapping, AIAA J., 17 (1979), 1281-1288.
doi: 10.2514/3.61308. |
[22] |
J. Heinhold and R. Albrecht,
Zur Praxis der konformen Abbildung, Rend. Circ. Mat. Palermo Ser. 2, 3 (1954), 130-148.
doi: 10.1007/BF02849374. |
[23] |
P. Henrici,
A general theory of osculation algorithms for conformal maps, J. Linear Alg. Appl., 52/53 (1983), 361-382.
doi: 10.1016/0024-3795(83)80024-X. |
[24] |
P. Henrici,
Applied and Computational Complex Analysis, John Wiley & Sons, Inc., New York, 1988. |
[25] |
H.-P. Hoidn,
Osculation methods for the conformal mapping of doubly connected regions, ZAMP, 33 (1982), 640-652.
|
[26] |
W. D. Hoskins and P. R. King,
Periodic cubic spline interpolation using parametric splines, The Computer Journal, 15 (1972), 282-283.
|
[27] |
R. M. James,
A general class of exact airfoil solutions, AIAA J., 9 (1972), 574-580.
doi: 10.2514/3.59038. |
[28] |
W. Koppenfels and F. Stallmann,
Praxis Der Konformen Abbildung, Springer, Berlin, 1959. |
[29] |
L. Landweber and T. Miloh,
Elimination of corners in the mapping of a closed curve, J. Engrg. Math., 6 (1972), 369-375.
doi: 10.1007/BF01535197. |
[30] |
R. S. Lehman,
Development of the mapping function at an analytic corner, Pacific J. Math., 7 (1957), 1437-1449.
doi: 10.2140/pjm.1957.7.1437. |
[31] |
H. Lewy,
Developments at the confluence of anaytic boundary conditions, Univ. of California Publ. in Math., 1 (1950), 247-280.
|
[32] |
D. E. Marshall,
Conformal welding for finitely connected regions, Comput. Methods Funct. Theory, 11 (2011), 655-669.
|
[33] |
J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, Sixth edition, Jones and bartlett, W, C. Brown, 2010. http://mathfaculty.fullerton.edu/mathews/complex.html |
[34] |
D. I. Meiron, S. A. Orszag and M. Israeli,
Applications of numerical conformal mapping, J. Comput. Phys., 40 (1981), 345-360.
doi: 10.1016/0021-9991(81)90215-1. |
[35] |
R. Menikoff and C. Zemach,
Methods for numerical conformal mapping, J. Comput. Phys., 36 (1980), 366-410.
doi: 10.1016/0021-9991(80)90166-7. |
[36] |
M. Nasser,
Fast computation of the circular map, Comput. Methods Funct. Theory, 15 (2015), 187-223.
doi: 10.1007/s40315-014-0098-3. |
[37] |
M. Nasser, T. Sakajo, A. Murid and L. K. Wei,
A fast computational method for potential flows in multiply connected coastal domains, Japan J. Indust. Appl. Math., 32 (2015), 205-236.
doi: 10.1007/s13160-015-0168-6. |
[38] |
S. T. O'Donnell and V. Rokhlin,
A fast algorithm for the numerical evaluation of conformal mappings, SIAM J. Sci. Statist. Comput., 10 (1989), 475-487.
doi: 10.1137/0910031. |
[39] |
N. Papamichael and N. Stylianopoulos,
Numerical Conformal Mapping - Domain Decomposition and the Mapping of Quadrilaterals, World Scientific, Singapore, 2010. |
[40] |
R. M. Porter,
An accelerated osculation method and its application to numerical conformal mapping, Complex Variables, 48 (2003), 569-582.
doi: 10.1080/0278107031000110892. |
[41] |
W. J. Prosnak,
Computation of Fluid Motions in Multiply Connected Domains, G. Braun, Karlsruhe, 1987. |
[42] |
W. J. Prosnak,
Conformal representation of arbitrary multiconnected airfoils, Bull. Acad. Pol. Sci., 25 (1977), 25-36 (591-602).
|
[43] |
R. Wegmann,
On Fornberg's numerical method for conformal mapping, SIAM J. Numer. Anal., 23 (1986), 1199-1213.
doi: 10.1137/0723081. |
[44] |
——, Methods for numerical conformal mapping, in Handbook of Complex Analysis, Geometric Function Theory, Vol. 2, (ed. R. Kuehnau), Elsevier, Amsterdam, (2005), 351-477. |
[45] |
B. R. Williams, An exact test case for the plane potential flow about two adjacent lifting airfoils,
RAE Technical Report No. 3717, London (1973). |


























N | Error |
64 | .1497 |
128 | .0697 |
256 | .0246 |
512 | .007 |
1024 | .0019 |
N | Error |
64 | .1497 |
128 | .0697 |
256 | .0246 |
512 | .007 |
1024 | .0019 |
Iteration | ||
1 | 6.0e-01 | 6.6e-01 |
2 | 3.3e-01 | 3.3e-01 |
3 | 7.0e-02 | 6.9e-02 |
4 | 2.6e-03 | 2.6e-03 |
5 | 4.0e-06 | 4.9e-06 |
6 | 4.2e-11 | 1.2e-06 |
7 | 0.0e+00 | 1.2e-06 |
8 | 0.0e+00 | 1.2e-06 |
Iteration | ||
1 | 6.0e-01 | 6.6e-01 |
2 | 3.3e-01 | 3.3e-01 |
3 | 7.0e-02 | 6.9e-02 |
4 | 2.6e-03 | 2.6e-03 |
5 | 4.0e-06 | 4.9e-06 |
6 | 4.2e-11 | 1.2e-06 |
7 | 0.0e+00 | 1.2e-06 |
8 | 0.0e+00 | 1.2e-06 |
Newton iterations | ||
1 | 1.6e+00 | 3.3e+00 |
2 | 7.5e-01 | 1.5e-00 |
3 | 1.3e-02 | 2.5e-02 |
4 | 5.5e-04 | 1.5e-03 |
5 | 5.6e-06 | 5.7e-06 |
6 | 3.7e-08 | 1.5e-08 |
7 | 3.4e-10 | 2.6e-11 |
8 | 2.2e-12 | 1.0e-13 |
9 | 3.1e-14 | 5.6e-14 |
10 | 1.8e-14 | 2.7e-14 |
Newton iterations | ||
1 | 1.6e+00 | 3.3e+00 |
2 | 7.5e-01 | 1.5e-00 |
3 | 1.3e-02 | 2.5e-02 |
4 | 5.5e-04 | 1.5e-03 |
5 | 5.6e-06 | 5.7e-06 |
6 | 3.7e-08 | 1.5e-08 |
7 | 3.4e-10 | 2.6e-11 |
8 | 2.2e-12 | 1.0e-13 |
9 | 3.1e-14 | 5.6e-14 |
10 | 1.8e-14 | 2.7e-14 |
trig | interp | spline | interp | |||
Ns | N | error | time | error | time | |
0.4 | 2000 | 256 | 1.2e-04 | 0.37 | 1.2e-04 | 0.06 |
2000 | 512 | 5.5e-06 | 0.89 | 5.5e-06 | 0.07 | |
2000 | 1024 | 2.3e-08 | 1.26 | 1.5e-07 | 0.07 | |
2000 | 2048 | 1.3e-09 | 2.59 | 2.3e-07 | 0.08 | |
2000 | 4096 | 1.3e-09 | 4.69 | 2.3e-07 | 0.11 | |
4096 | 4096 | 6.9e-14 | 10.09 | 9.4e-09 | 0.13 | |
0.2 | 2000 | 256 | 1.0e-04 | 0.43 | 1.0e-04 | 0.07 |
2000 | 512 | 2.0e-06 | 0.84 | 1.7e-06 | 0.08 | |
2000 | 1024 | 1.5e-06 | 1.25 | 6.0e-07 | 0.08 | |
2000 | 2048 | 1.3e-06 | 2.55 | 5.7e-07 | 0.09 | |
2000 | 4096 | 1.2e-06 | 4.48 | 5.7e-07 | 0.12 | |
4096 | 4096 | 3.4e-09 | 12.60 | 3.4e-08 | 0.16 | |
0.1 | 2000 | 256 | 2.4e-03 | 0.39 | 2.5e-03 | 0.07 |
2000 | 512 | 2.3e-04 | 0.74 | 2.3e-04 | 0.08 | |
2000 | 1024 | 7.2e-05 | 1.49 | 4.4e-06 | 0.08 | |
2000 | 2048 | 7.9e-05 | 2.77 | 1.3e-06 | 0.09 | |
2000 | 4096 | 6.6e-05 | 5.46 | 1.3e-06 | 0.15 | |
4096 | 4096 | 7.9e-06 | 10.13 | 3.6e-07 | 0.16 |
trig | interp | spline | interp | |||
Ns | N | error | time | error | time | |
0.4 | 2000 | 256 | 1.2e-04 | 0.37 | 1.2e-04 | 0.06 |
2000 | 512 | 5.5e-06 | 0.89 | 5.5e-06 | 0.07 | |
2000 | 1024 | 2.3e-08 | 1.26 | 1.5e-07 | 0.07 | |
2000 | 2048 | 1.3e-09 | 2.59 | 2.3e-07 | 0.08 | |
2000 | 4096 | 1.3e-09 | 4.69 | 2.3e-07 | 0.11 | |
4096 | 4096 | 6.9e-14 | 10.09 | 9.4e-09 | 0.13 | |
0.2 | 2000 | 256 | 1.0e-04 | 0.43 | 1.0e-04 | 0.07 |
2000 | 512 | 2.0e-06 | 0.84 | 1.7e-06 | 0.08 | |
2000 | 1024 | 1.5e-06 | 1.25 | 6.0e-07 | 0.08 | |
2000 | 2048 | 1.3e-06 | 2.55 | 5.7e-07 | 0.09 | |
2000 | 4096 | 1.2e-06 | 4.48 | 5.7e-07 | 0.12 | |
4096 | 4096 | 3.4e-09 | 12.60 | 3.4e-08 | 0.16 | |
0.1 | 2000 | 256 | 2.4e-03 | 0.39 | 2.5e-03 | 0.07 |
2000 | 512 | 2.3e-04 | 0.74 | 2.3e-04 | 0.08 | |
2000 | 1024 | 7.2e-05 | 1.49 | 4.4e-06 | 0.08 | |
2000 | 2048 | 7.9e-05 | 2.77 | 1.3e-06 | 0.09 | |
2000 | 4096 | 6.6e-05 | 5.46 | 1.3e-06 | 0.15 | |
4096 | 4096 | 7.9e-06 | 10.13 | 3.6e-07 | 0.16 |
ig | ig | ig | ig | ||||
0 | .800 | 0 | .400 | 0 | .200 | 0 | .100 |
c | .804 | k | .416 | k | .239 | k | .150 |
c | .950 | k | .571 | k | .449 | k | .361 |
c | .594 | k | .464 | k | .388 | ||
c | .770 | k | .535 | k | .484 | ||
c | .775 | c | .563 | c | .528 | ||
c | .912 | c | .717 | c | .704 | ||
c | .912 | c | .725 | c | .713 | ||
c | .961 | c | .885 | c | .917 | ||
c | .962 | c | .886 | c | .918 | ||
c | .968 | c | .957 | c | .960 |
ig | ig | ig | ig | ||||
0 | .800 | 0 | .400 | 0 | .200 | 0 | .100 |
c | .804 | k | .416 | k | .239 | k | .150 |
c | .950 | k | .571 | k | .449 | k | .361 |
c | .594 | k | .464 | k | .388 | ||
c | .770 | k | .535 | k | .484 | ||
c | .775 | c | .563 | c | .528 | ||
c | .912 | c | .717 | c | .704 | ||
c | .912 | c | .725 | c | .713 | ||
c | .961 | c | .885 | c | .917 | ||
c | .962 | c | .886 | c | .918 | ||
c | .968 | c | .957 | c | .960 |
Ns | N | time (m=4) | time (m=2) |
200 | 128 | 1.45 | 0.77 |
200 | 256 | 3.65 | 1.64 |
200 | 512 | 15.92 | 3.36 |
400 | 128 | 1.71 | 0.73 |
400 | 256 | 3.54 | 1.11 |
400 | 512 | 15.66 | 3.28 |
800 | 128 | 1.56 | 0.81 |
800 | 256 | 3.62 | 1.16 |
800 | 512 | 15.73 | 3.30 |
Ns | N | time (m=4) | time (m=2) |
200 | 128 | 1.45 | 0.77 |
200 | 256 | 3.65 | 1.64 |
200 | 512 | 15.92 | 3.36 |
400 | 128 | 1.71 | 0.73 |
400 | 256 | 3.54 | 1.11 |
400 | 512 | 15.66 | 3.28 |
800 | 128 | 1.56 | 0.81 |
800 | 256 | 3.62 | 1.16 |
800 | 512 | 15.73 | 3.30 |
Newton iterations | 3 rectangles |
Joukowski airfoil |
Two cosine airfoils |
1 | 2.3e+01 | 1.0e+00 | 3.4e+00 |
2 | 2.3e+01 | 3.0e-01 | 3.0e-01 |
3 | 3.1e+00 | 2.3e-02 | 3.0e-03 |
4 | 1.0e+00 | 1.2e-04 | 1.6e-05 |
5 | 5.2e-01 | 2.1e-09 | 3.4e-08 |
6 | 6.4e-02 | 5.4e-10 | 1.1e-10 |
7 | 6.9e-03 | 5.4e-10 | 3.3e-13 |
8 | 1.2e-04 | 5.4e-10 | 1.7e-15 |
9 | 1.9e-06 | 5.4e-10 | 4.1e-15 |
10 | 4.0e-08 | 5.4e-10 | 9.7e-15 |
11 | 6.2e-10 | ||
12 | 1.3e-11 | ||
13 | 1.8e-13 | ||
14 | 1.6e-14 |
Newton iterations | 3 rectangles |
Joukowski airfoil |
Two cosine airfoils |
1 | 2.3e+01 | 1.0e+00 | 3.4e+00 |
2 | 2.3e+01 | 3.0e-01 | 3.0e-01 |
3 | 3.1e+00 | 2.3e-02 | 3.0e-03 |
4 | 1.0e+00 | 1.2e-04 | 1.6e-05 |
5 | 5.2e-01 | 2.1e-09 | 3.4e-08 |
6 | 6.4e-02 | 5.4e-10 | 1.1e-10 |
7 | 6.9e-03 | 5.4e-10 | 3.3e-13 |
8 | 1.2e-04 | 5.4e-10 | 1.7e-15 |
9 | 1.9e-06 | 5.4e-10 | 4.1e-15 |
10 | 4.0e-08 | 5.4e-10 | 9.7e-15 |
11 | 6.2e-10 | ||
12 | 1.3e-11 | ||
13 | 1.8e-13 | ||
14 | 1.6e-14 |
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