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Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems
Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182, USA |
In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the first-and second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p+3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.
References:
[1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. |
[2] |
S. Adjerid and M. Baccouch,
A superconvergent local discontinuous Galerkin method for elliptic problems, Journal of Scientific Computing, 52 (2012), 113-152.
doi: 10.1007/s10915-011-9537-8. |
[3] |
S. Adjerid and D. Issaev, Superconvergence of the local discontinuous Galerkin method applied to diffusion problems, in: K. Bathe (ed. ), Proceedings of the Third MIT Conference on Computational Fluid and Solid Mechanics, vol. 3, Elsevier, 2005. |
[4] |
S. Adjerid and A. Klauser,
Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems, Journal of Scientific Computing, 22 (2005), 5-24.
doi: 10.1007/s10915-004-4133-9. |
[5] |
M. Baccouch,
A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209/212 (2012), 129-143.
doi: 10.1016/j.cma.2011.10.012. |
[6] |
M. Baccouch,
Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems, Applied Mathematics and Computation, 226 (2014), 455-483.
doi: 10.1016/j.amc.2013.10.026. |
[7] |
M. Baccouch,
Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension, Computers & Mathematics with Applications, 67 (2014), 1130-1153.
doi: 10.1016/j.camwa.2013.12.014. |
[8] |
M. Baccouch,
A superconvergent local discontinuous Galerkin method for the second-order wave equation on cartesian grids, Computers and Mathematics with Applications, 68 (2014), 1250-1278.
doi: 10.1016/j.camwa.2014.08.023. |
[9] |
M. Baccouch,
Asymptotically exact local discontinuous Galerkin error estimates for the linearized Korteweg-de Vries equation in one space dimension, International Journal of Numerical Analysis and Modeling, 12 (2015), 162-195.
|
[10] |
M. Baccouch,
Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension, Journal of Computational Mathematics, 34 (2016), 511-531.
doi: 10.4208/jcm.1603-m2015-0317. |
[11] |
T. Benjamin, J. Bona and J. Mahony,
Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.
doi: 10.1098/rsta.1972.0032. |
[12] |
J. L. Bona, H. Chen, O. Karakashian and Y. Xing,
Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Mathematics of Computation, 82 (2013), 1401-1432.
doi: 10.1090/S0025-5718-2013-02661-0. |
[13] |
W. Cao and Z. Zhang,
Superconvergence of local discontinuous Galerkin methods for one-dimensional linear parabolic equations, Mathematics of Computation, 85 (2016), 63-84.
|
[14] |
P. Castillo,
A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 4675-4685.
doi: 10.1016/S0045-7825(03)00445-6. |
[15] |
P. Castillo,
A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, 56 (2006), 1307-1313.
doi: 10.1016/j.apnum.2006.03.016. |
[16] |
P. Castillo, B. Cockburn, D. Schötzau and C. Schwab,
Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Mathematic of Computation, 71 (2002), 455-478.
|
[17] |
F. Celiker and B. Cockburn,
Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Mathematics of Computation, 76 (2007), 67-96.
doi: 10.1090/S0025-5718-06-01895-3. |
[18] |
Y. Cheng and C.-W. Shu,
Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), 4044-4072.
doi: 10.1137/090747701. |
[19] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978. |
[20] |
B. Cockburn, G. Kanschat and D. Schötzau,
A locally conservative LDG method for the incompressible Navier-Stokes equations, Mathematics of Compuatation, 74 (2005), 1067-1095.
|
[21] |
B. Cockburn, G. Kanschat and D. Schötzau,
The local discontinuous Galerkin method for linearized incompressible fluid flow: A review, Computers & Fluids, 34 (2005), 491-506.
doi: 10.1016/j.compfluid.2003.08.005. |
[22] |
B. Cockburn, G. E. Karniadakis and C. W. Shu, Discontinuous Galerkin Methods Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000. |
[23] |
B. Cockburn and C. W. Shu,
TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws Ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435.
|
[24] |
B. Cockburn and C. W. Shu,
The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463.
doi: 10.1137/S0036142997316712. |
[25] |
T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2010. |
[26] |
K. D. Devine and J. E. Flaherty,
Parallel adaptive hp-refinement techniques for conservation laws, Computer Methods in Applied Mechanics and Engineering, 20 (1996), 367-386.
doi: 10.1016/0168-9274(95)00103-4. |
[27] |
J. E. Flaherty, R. Loy, M. S. Shephard, B. K. Szymanski, J. D. Teresco and L. H. Ziantz,
Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws, Journal of Parallel and Distributed Computing, 47 (1997), 139-152.
doi: 10.1006/jpdc.1997.1412. |
[28] |
C. Hufford and Y. Xing,
Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation, Journal of Computational and Applied Mathematics, 255 (2014), 441-455.
doi: 10.1016/j.cam.2013.06.004. |
[29] |
V. Kucera,
On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA Journal of Numerical Analysis, 34 (2014), 820-861.
doi: 10.1093/imanum/drt007. |
[30] |
X. Meng, C.-W. Shu, Q. Zhang and B. Wu,
Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM Journal on Numerical Analysis, 50 (2012), 2336-2356.
doi: 10.1137/110857635. |
[31] |
S. Osher,
Riemann solvers, the entropy condition, and difference approximations, SIAM Journal on Numerical Analysis, 21 (1984), 217-235.
doi: 10.1137/0721016. |
[32] |
T. E. Peterson,
A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis, 28 (1991), 133-140.
doi: 10.1137/0728006. |
[33] |
B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008. |
[34] |
A. Samii, N. Panda and C. Craig Michoskiand Dawson,
A hybridized discontinuous galerkin method for the nonlinear Korteweg-de Vries equation, Journal of Scientific Computing, 68 (2016), 191-212.
doi: 10.1007/s10915-015-0133-1. |
[35] |
D. Schötzau and C. Schwab,
Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method, SIAM Journal on Numerical Analysis, 38 (2000), 837-875.
doi: 10.1137/S0036142999352394. |
[36] |
H. Wang, C.-W. Shu and Q. Zhang,
Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems, Applied Mathematics and Computation, 272 (2016), 237-258, recent Advances in Numerical Methods for Hyperbolic Partial Differential Equations.
doi: 10.1016/j.amc.2015.02.067. |
[37] |
Y. Xing, C.-S. Chou and C.-W. Shu,
Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problems and Imaging, 7 (2013), 967-986.
doi: 10.3934/ipi.2013.7.967. |
[38] |
Y. Xu and C.-W. Shu,
Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3805-3822.
doi: 10.1016/j.cma.2006.10.043. |
[39] |
Y. Xu and C.-W. Shu,
Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Communications in Computational Physics, 7 (2010), 1-46.
|
[40] |
Y. Xu and C.-W. Shu,
Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations, SIAM Journal on Numerical Analysis, 50 (2012), 79-104.
doi: 10.1137/11082258X. |
[41] |
J. Yan and C.-W. Shu,
A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, 40 (2002), 769-791.
doi: 10.1137/S0036142901390378. |
[42] |
Y. Yang and C.-W. Shu,
Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM Journal on Numerical Analysis, 50 (2012), 3110-3133.
doi: 10.1137/110857647. |
[43] |
Y. Yang and C.-W. Shu,
Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations, Journal of Computational Mathematics, 33 (2015), 323-340.
doi: 10.4208/jcm.1502-m2014-0001. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. |
[2] |
S. Adjerid and M. Baccouch,
A superconvergent local discontinuous Galerkin method for elliptic problems, Journal of Scientific Computing, 52 (2012), 113-152.
doi: 10.1007/s10915-011-9537-8. |
[3] |
S. Adjerid and D. Issaev, Superconvergence of the local discontinuous Galerkin method applied to diffusion problems, in: K. Bathe (ed. ), Proceedings of the Third MIT Conference on Computational Fluid and Solid Mechanics, vol. 3, Elsevier, 2005. |
[4] |
S. Adjerid and A. Klauser,
Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems, Journal of Scientific Computing, 22 (2005), 5-24.
doi: 10.1007/s10915-004-4133-9. |
[5] |
M. Baccouch,
A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209/212 (2012), 129-143.
doi: 10.1016/j.cma.2011.10.012. |
[6] |
M. Baccouch,
Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems, Applied Mathematics and Computation, 226 (2014), 455-483.
doi: 10.1016/j.amc.2013.10.026. |
[7] |
M. Baccouch,
Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension, Computers & Mathematics with Applications, 67 (2014), 1130-1153.
doi: 10.1016/j.camwa.2013.12.014. |
[8] |
M. Baccouch,
A superconvergent local discontinuous Galerkin method for the second-order wave equation on cartesian grids, Computers and Mathematics with Applications, 68 (2014), 1250-1278.
doi: 10.1016/j.camwa.2014.08.023. |
[9] |
M. Baccouch,
Asymptotically exact local discontinuous Galerkin error estimates for the linearized Korteweg-de Vries equation in one space dimension, International Journal of Numerical Analysis and Modeling, 12 (2015), 162-195.
|
[10] |
M. Baccouch,
Optimal a posteriori error estimates of the local discontinuous Galerkin method for convection-diffusion problems in one space dimension, Journal of Computational Mathematics, 34 (2016), 511-531.
doi: 10.4208/jcm.1603-m2015-0317. |
[11] |
T. Benjamin, J. Bona and J. Mahony,
Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.
doi: 10.1098/rsta.1972.0032. |
[12] |
J. L. Bona, H. Chen, O. Karakashian and Y. Xing,
Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Mathematics of Computation, 82 (2013), 1401-1432.
doi: 10.1090/S0025-5718-2013-02661-0. |
[13] |
W. Cao and Z. Zhang,
Superconvergence of local discontinuous Galerkin methods for one-dimensional linear parabolic equations, Mathematics of Computation, 85 (2016), 63-84.
|
[14] |
P. Castillo,
A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 4675-4685.
doi: 10.1016/S0045-7825(03)00445-6. |
[15] |
P. Castillo,
A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, 56 (2006), 1307-1313.
doi: 10.1016/j.apnum.2006.03.016. |
[16] |
P. Castillo, B. Cockburn, D. Schötzau and C. Schwab,
Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Mathematic of Computation, 71 (2002), 455-478.
|
[17] |
F. Celiker and B. Cockburn,
Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Mathematics of Computation, 76 (2007), 67-96.
doi: 10.1090/S0025-5718-06-01895-3. |
[18] |
Y. Cheng and C.-W. Shu,
Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), 4044-4072.
doi: 10.1137/090747701. |
[19] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978. |
[20] |
B. Cockburn, G. Kanschat and D. Schötzau,
A locally conservative LDG method for the incompressible Navier-Stokes equations, Mathematics of Compuatation, 74 (2005), 1067-1095.
|
[21] |
B. Cockburn, G. Kanschat and D. Schötzau,
The local discontinuous Galerkin method for linearized incompressible fluid flow: A review, Computers & Fluids, 34 (2005), 491-506.
doi: 10.1016/j.compfluid.2003.08.005. |
[22] |
B. Cockburn, G. E. Karniadakis and C. W. Shu, Discontinuous Galerkin Methods Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000. |
[23] |
B. Cockburn and C. W. Shu,
TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws Ⅱ: General framework, Mathematics of Computation, 52 (1989), 411-435.
|
[24] |
B. Cockburn and C. W. Shu,
The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), 2440-2463.
doi: 10.1137/S0036142997316712. |
[25] |
T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2010. |
[26] |
K. D. Devine and J. E. Flaherty,
Parallel adaptive hp-refinement techniques for conservation laws, Computer Methods in Applied Mechanics and Engineering, 20 (1996), 367-386.
doi: 10.1016/0168-9274(95)00103-4. |
[27] |
J. E. Flaherty, R. Loy, M. S. Shephard, B. K. Szymanski, J. D. Teresco and L. H. Ziantz,
Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws, Journal of Parallel and Distributed Computing, 47 (1997), 139-152.
doi: 10.1006/jpdc.1997.1412. |
[28] |
C. Hufford and Y. Xing,
Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation, Journal of Computational and Applied Mathematics, 255 (2014), 441-455.
doi: 10.1016/j.cam.2013.06.004. |
[29] |
V. Kucera,
On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA Journal of Numerical Analysis, 34 (2014), 820-861.
doi: 10.1093/imanum/drt007. |
[30] |
X. Meng, C.-W. Shu, Q. Zhang and B. Wu,
Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM Journal on Numerical Analysis, 50 (2012), 2336-2356.
doi: 10.1137/110857635. |
[31] |
S. Osher,
Riemann solvers, the entropy condition, and difference approximations, SIAM Journal on Numerical Analysis, 21 (1984), 217-235.
doi: 10.1137/0721016. |
[32] |
T. E. Peterson,
A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis, 28 (1991), 133-140.
doi: 10.1137/0728006. |
[33] |
B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008. |
[34] |
A. Samii, N. Panda and C. Craig Michoskiand Dawson,
A hybridized discontinuous galerkin method for the nonlinear Korteweg-de Vries equation, Journal of Scientific Computing, 68 (2016), 191-212.
doi: 10.1007/s10915-015-0133-1. |
[35] |
D. Schötzau and C. Schwab,
Time discretization of parabolic problems by the $hp$-version of the discontinuous Galerkin finite element method, SIAM Journal on Numerical Analysis, 38 (2000), 837-875.
doi: 10.1137/S0036142999352394. |
[36] |
H. Wang, C.-W. Shu and Q. Zhang,
Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems, Applied Mathematics and Computation, 272 (2016), 237-258, recent Advances in Numerical Methods for Hyperbolic Partial Differential Equations.
doi: 10.1016/j.amc.2015.02.067. |
[37] |
Y. Xing, C.-S. Chou and C.-W. Shu,
Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Problems and Imaging, 7 (2013), 967-986.
doi: 10.3934/ipi.2013.7.967. |
[38] |
Y. Xu and C.-W. Shu,
Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3805-3822.
doi: 10.1016/j.cma.2006.10.043. |
[39] |
Y. Xu and C.-W. Shu,
Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Communications in Computational Physics, 7 (2010), 1-46.
|
[40] |
Y. Xu and C.-W. Shu,
Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations, SIAM Journal on Numerical Analysis, 50 (2012), 79-104.
doi: 10.1137/11082258X. |
[41] |
J. Yan and C.-W. Shu,
A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, 40 (2002), 769-791.
doi: 10.1137/S0036142901390378. |
[42] |
Y. Yang and C.-W. Shu,
Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM Journal on Numerical Analysis, 50 (2012), 3110-3133.
doi: 10.1137/110857647. |
[43] |
Y. Yang and C.-W. Shu,
Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations, Journal of Computational Mathematics, 33 (2015), 323-340.
doi: 10.4208/jcm.1502-m2014-0001. |


p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 4.5846e-02 | 1.8733e-03 | 3.0396e-05 | 9.4387e-07 | ||||
20 | 1.0865e-02 | 2.0771 | 1.1247e-04 | 4.0580 | 1.8971e-06 | 4.0020 | 2.9595e-08 | 4.9952 |
30 | 4.7435e-03 | 2.0440 | 3.0068e-05 | 3.2536 | 3.7529e-07 | 3.9964 | 3.9006e-09 | 4.9979 |
40 | 2.6395e-03 | 2.0376 | 1.2436e-05 | 3.0689 | 1.1883e-07 | 3.9975 | 9.2598e-10 | 4.9987 |
50 | 1.6794e-03 | 2.0263 | 6.3325e-06 | 3.0245 | 4.8695e-08 | 3.9980 | 3.0349e-10 | 4.9990 |
60 | 1.1623e-03 | 2.0186 | 3.6576e-06 | 3.0105 | 2.3490e-08 | 3.9984 | 1.2198e-10 | 4.9994 |
70 | 8.5246e-04 | 2.0112 | 2.3015e-06 | 3.0052 | 1.2682e-08 | 3.9986 | 5.6441e-11 | 4.9994 |
80 | 6.5263e-04 | 2.0004 | 1.5413e-06 | 3.0026 | 7.4350e-09 | 3.9989 | 2.8952e-11 | 4.9992 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 4.5846e-02 | 1.8733e-03 | 3.0396e-05 | 9.4387e-07 | ||||
20 | 1.0865e-02 | 2.0771 | 1.1247e-04 | 4.0580 | 1.8971e-06 | 4.0020 | 2.9595e-08 | 4.9952 |
30 | 4.7435e-03 | 2.0440 | 3.0068e-05 | 3.2536 | 3.7529e-07 | 3.9964 | 3.9006e-09 | 4.9979 |
40 | 2.6395e-03 | 2.0376 | 1.2436e-05 | 3.0689 | 1.1883e-07 | 3.9975 | 9.2598e-10 | 4.9987 |
50 | 1.6794e-03 | 2.0263 | 6.3325e-06 | 3.0245 | 4.8695e-08 | 3.9980 | 3.0349e-10 | 4.9990 |
60 | 1.1623e-03 | 2.0186 | 3.6576e-06 | 3.0105 | 2.3490e-08 | 3.9984 | 1.2198e-10 | 4.9994 |
70 | 8.5246e-04 | 2.0112 | 2.3015e-06 | 3.0052 | 1.2682e-08 | 3.9986 | 5.6441e-11 | 4.9994 |
80 | 6.5263e-04 | 2.0004 | 1.5413e-06 | 3.0026 | 7.4350e-09 | 3.9989 | 2.8952e-11 | 4.9992 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 4.0632e-02 | 2.1357e-03 | 8.1067e-05 | 2.5683e-06 | ||||
20 | 1.0377e-02 | 1.9692 | 2.6569e-04 | 3.0069 | 5.1201e-06 | 3.9849 | 8.0223e-08 | 5.0007 |
30 | 4.6494e-03 | 1.9801 | 7.9028e-05 | 2.9905 | 1.0154e-06 | 3.9902 | 1.0585e-08 | 4.9952 |
40 | 2.6263e-03 | 1.9854 | 3.3377e-05 | 2.9961 | 3.2190e-07 | 3.9933 | 2.5136e-09 | 4.9976 |
50 | 1.6852e-03 | 1.9884 | 1.7106e-05 | 2.9956 | 1.3200e-07 | 3.9949 | 8.2408e-10 | 4.9977 |
60 | 1.1723e-03 | 1.9905 | 9.9056e-06 | 2.9965 | 6.3705e-08 | 3.9959 | 3.3129e-10 | 4.9982 |
70 | 8.6237e-04 | 1.9918 | 6.2400e-06 | 2.9978 | 3.4405e-08 | 3.9965 | 1.5331e-10 | 4.9986 |
80 | 6.6088e-04 | 1.9929 | 4.1817e-06 | 2.9975 | 2.0176e-08 | 3.9969 | 7.8645e-11 | 4.9990 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 4.0632e-02 | 2.1357e-03 | 8.1067e-05 | 2.5683e-06 | ||||
20 | 1.0377e-02 | 1.9692 | 2.6569e-04 | 3.0069 | 5.1201e-06 | 3.9849 | 8.0223e-08 | 5.0007 |
30 | 4.6494e-03 | 1.9801 | 7.9028e-05 | 2.9905 | 1.0154e-06 | 3.9902 | 1.0585e-08 | 4.9952 |
40 | 2.6263e-03 | 1.9854 | 3.3377e-05 | 2.9961 | 3.2190e-07 | 3.9933 | 2.5136e-09 | 4.9976 |
50 | 1.6852e-03 | 1.9884 | 1.7106e-05 | 2.9956 | 1.3200e-07 | 3.9949 | 8.2408e-10 | 4.9977 |
60 | 1.1723e-03 | 1.9905 | 9.9056e-06 | 2.9965 | 6.3705e-08 | 3.9959 | 3.3129e-10 | 4.9982 |
70 | 8.6237e-04 | 1.9918 | 6.2400e-06 | 2.9978 | 3.4405e-08 | 3.9965 | 1.5331e-10 | 4.9986 |
80 | 6.6088e-04 | 1.9929 | 4.1817e-06 | 2.9975 | 2.0176e-08 | 3.9969 | 7.8645e-11 | 4.9990 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 5.6489e-02 | 2.6049e-03 | 1.0690e-04 | 3.1989e-06 | ||||
20 | 1.4417e-02 | 1.9702 | 3.1480e-04 | 3.0487 | 6.8578e-06 | 3.9624 | 9.3967e-08 | 5.0893 |
30 | 6.4714e-03 | 1.9755 | 9.2291e-05 | 3.0261 | 1.3672e-06 | 3.9772 | 1.2251e-08 | 5.0247 |
40 | 3.6555e-03 | 1.9854 | 3.8721e-05 | 3.0192 | 4.3448e-07 | 3.9849 | 2.8903e-09 | 5.0203 |
50 | 2.3462e-03 | 1.9872 | 1.9756e-05 | 3.0157 | 1.7839e-07 | 3.9893 | 9.4370e-10 | 5.0161 |
60 | 1.6322e-03 | 1.9903 | 1.1409e-05 | 3.0115 | 8.6173e-08 | 3.9908 | 3.7854e-10 | 5.0103 |
70 | 1.2006e-03 | 1.9923 | 7.1733e-06 | 3.0103 | 4.6565e-08 | 3.9929 | 1.7483e-10 | 5.0114 |
80 | 9.2012e-04 | 1.9926 | 4.8003e-06 | 3.0082 | 2.7319e-08 | 3.9936 | 8.9582e-11 | 5.0075 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 5.6489e-02 | 2.6049e-03 | 1.0690e-04 | 3.1989e-06 | ||||
20 | 1.4417e-02 | 1.9702 | 3.1480e-04 | 3.0487 | 6.8578e-06 | 3.9624 | 9.3967e-08 | 5.0893 |
30 | 6.4714e-03 | 1.9755 | 9.2291e-05 | 3.0261 | 1.3672e-06 | 3.9772 | 1.2251e-08 | 5.0247 |
40 | 3.6555e-03 | 1.9854 | 3.8721e-05 | 3.0192 | 4.3448e-07 | 3.9849 | 2.8903e-09 | 5.0203 |
50 | 2.3462e-03 | 1.9872 | 1.9756e-05 | 3.0157 | 1.7839e-07 | 3.9893 | 9.4370e-10 | 5.0161 |
60 | 1.6322e-03 | 1.9903 | 1.1409e-05 | 3.0115 | 8.6173e-08 | 3.9908 | 3.7854e-10 | 5.0103 |
70 | 1.2006e-03 | 1.9923 | 7.1733e-06 | 3.0103 | 4.6565e-08 | 3.9929 | 1.7483e-10 | 5.0114 |
80 | 9.2012e-04 | 1.9926 | 4.8003e-06 | 3.0082 | 2.7319e-08 | 3.9936 | 8.9582e-11 | 5.0075 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 8.2164e-03 | 2.8426e-04 | 8.5926e-06 | 2.7880e-07 | ||||
20 | 9.2680e-04 | 3.1482 | 1.4330e-05 | 4.3101 | 2.1723e-07 | 5.3058 | 2.5301e-09 | 6.7839 |
30 | 2.7332e-04 | 3.0116 | 2.7906e-06 | 4.0351 | 2.8408e-08 | 5.0172 | 2.2288e-10 | 5.9916 |
40 | 1.1432e-04 | 3.0299 | 8.4343e-07 | 4.1592 | 6.6969e-09 | 5.0230 | 3.4499e-11 | 6.4853 |
50 | 5.8315e-05 | 3.0166 | 3.4528e-07 | 4.0025 | 2.1868e-09 | 5.0156 | 9.0621e-12 | 5.9909 |
60 | 3.3724e-05 | 3.0038 | 1.6466e-07 | 4.0614 | 8.7923e-10 | 4.9975 | 2.9629e-12 | 6.1317 |
70 | 2.1167e-05 | 3.0215 | 8.7954e-08 | 4.0679 | 4.0603e-10 | 5.0121 | 1.1105e-12 | 6.3662 |
80 | 1.4176e-05 | 3.0022 | 5.1484e-08 | 4.0106 | 2.0829e-10 | 4.9988 | 4.9946e-13 | 5.9839 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 8.2164e-03 | 2.8426e-04 | 8.5926e-06 | 2.7880e-07 | ||||
20 | 9.2680e-04 | 3.1482 | 1.4330e-05 | 4.3101 | 2.1723e-07 | 5.3058 | 2.5301e-09 | 6.7839 |
30 | 2.7332e-04 | 3.0116 | 2.7906e-06 | 4.0351 | 2.8408e-08 | 5.0172 | 2.2288e-10 | 5.9916 |
40 | 1.1432e-04 | 3.0299 | 8.4343e-07 | 4.1592 | 6.6969e-09 | 5.0230 | 3.4499e-11 | 6.4853 |
50 | 5.8315e-05 | 3.0166 | 3.4528e-07 | 4.0025 | 2.1868e-09 | 5.0156 | 9.0621e-12 | 5.9909 |
60 | 3.3724e-05 | 3.0038 | 1.6466e-07 | 4.0614 | 8.7923e-10 | 4.9975 | 2.9629e-12 | 6.1317 |
70 | 2.1167e-05 | 3.0215 | 8.7954e-08 | 4.0679 | 4.0603e-10 | 5.0121 | 1.1105e-12 | 6.3662 |
80 | 1.4176e-05 | 3.0022 | 5.1484e-08 | 4.0106 | 2.0829e-10 | 4.9988 | 4.9946e-13 | 5.9839 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 6.2562e-02 | 9.8763e-04 | 3.1041e-05 | 9.8565e-07 | ||||
20 | 1.5553e-02 | 2.0081 | 1.0290e-04 | 3.2627 | 1.9775e-06 | 3.9724 | 3.0496e-08 | 5.0144 |
30 | 6.8955e-03 | 2.0061 | 3.0045e-05 | 3.0362 | 3.9293e-07 | 3.9854 | 4.0031e-09 | 5.0079 |
40 | 3.8742e-03 | 2.0040 | 1.2632e-05 | 3.0119 | 1.2468e-07 | 3.9901 | 9.4846e-10 | 5.0055 |
50 | 2.4781e-03 | 2.0025 | 6.4589e-06 | 3.0060 | 5.1154e-08 | 3.9925 | 3.1049e-10 | 5.0043 |
60 | 1.7203e-03 | 2.0021 | 3.7351e-06 | 3.0039 | 2.4696e-08 | 3.9940 | 1.2470e-10 | 5.0035 |
70 | 1.2636e-03 | 2.0014 | 2.3511e-06 | 3.0028 | 1.3341e-08 | 3.9948 | 5.7668e-11 | 5.0030 |
80 | 9.6732e-04 | 2.0009 | 1.5745e-06 | 3.0026 | 7.8247e-09 | 3.9957 | 2.9568e-11 | 5.0026 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 6.2562e-02 | 9.8763e-04 | 3.1041e-05 | 9.8565e-07 | ||||
20 | 1.5553e-02 | 2.0081 | 1.0290e-04 | 3.2627 | 1.9775e-06 | 3.9724 | 3.0496e-08 | 5.0144 |
30 | 6.8955e-03 | 2.0061 | 3.0045e-05 | 3.0362 | 3.9293e-07 | 3.9854 | 4.0031e-09 | 5.0079 |
40 | 3.8742e-03 | 2.0040 | 1.2632e-05 | 3.0119 | 1.2468e-07 | 3.9901 | 9.4846e-10 | 5.0055 |
50 | 2.4781e-03 | 2.0025 | 6.4589e-06 | 3.0060 | 5.1154e-08 | 3.9925 | 3.1049e-10 | 5.0043 |
60 | 1.7203e-03 | 2.0021 | 3.7351e-06 | 3.0039 | 2.4696e-08 | 3.9940 | 1.2470e-10 | 5.0035 |
70 | 1.2636e-03 | 2.0014 | 2.3511e-06 | 3.0028 | 1.3341e-08 | 3.9948 | 5.7668e-11 | 5.0030 |
80 | 9.6732e-04 | 2.0009 | 1.5745e-06 | 3.0026 | 7.8247e-09 | 3.9957 | 2.9568e-11 | 5.0026 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 6.9746e-02 | 5.7247e-03 | 4.8429e-04 | 3.8023e-05 | ||||
20 | 1.7340e-02 | 2.0080 | 7.3031e-04 | 2.9706 | 3.0517e-05 | 3.9882 | 1.2060e-06 | 4.9786 |
30 | 7.6873e-03 | 2.0062 | 2.1756e-04 | 2.9867 | 6.0452e-06 | 3.9930 | 1.5947e-07 | 4.9898 |
40 | 4.3192e-03 | 2.0039 | 9.1951e-05 | 2.9936 | 1.9150e-06 | 3.9959 | 3.7902e-08 | 4.9946 |
50 | 2.7627e-03 | 2.0026 | 4.7116e-05 | 2.9965 | 7.8488e-07 | 3.9972 | 1.2429e-08 | 4.9967 |
60 | 1.9179e-03 | 2.0018 | 2.7278e-05 | 2.9976 | 3.7864e-07 | 3.9981 | 4.9968e-09 | 4.9980 |
70 | 1.4087e-03 | 2.0017 | 1.7182e-05 | 2.9985 | 2.0443e-07 | 3.9984 | 2.3124e-09 | 4.9984 |
80 | 1.0784e-03 | 2.0009 | 1.1512e-05 | 2.9991 | 1.1985e-07 | 3.9989 | 1.1862e-09 | 4.9991 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 6.9746e-02 | 5.7247e-03 | 4.8429e-04 | 3.8023e-05 | ||||
20 | 1.7340e-02 | 2.0080 | 7.3031e-04 | 2.9706 | 3.0517e-05 | 3.9882 | 1.2060e-06 | 4.9786 |
30 | 7.6873e-03 | 2.0062 | 2.1756e-04 | 2.9867 | 6.0452e-06 | 3.9930 | 1.5947e-07 | 4.9898 |
40 | 4.3192e-03 | 2.0039 | 9.1951e-05 | 2.9936 | 1.9150e-06 | 3.9959 | 3.7902e-08 | 4.9946 |
50 | 2.7627e-03 | 2.0026 | 4.7116e-05 | 2.9965 | 7.8488e-07 | 3.9972 | 1.2429e-08 | 4.9967 |
60 | 1.9179e-03 | 2.0018 | 2.7278e-05 | 2.9976 | 3.7864e-07 | 3.9981 | 4.9968e-09 | 4.9980 |
70 | 1.4087e-03 | 2.0017 | 1.7182e-05 | 2.9985 | 2.0443e-07 | 3.9984 | 2.3124e-09 | 4.9984 |
80 | 1.0784e-03 | 2.0009 | 1.1512e-05 | 2.9991 | 1.1985e-07 | 3.9989 | 1.1862e-09 | 4.9991 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 8.1901e-02 | 1.0758e-02 | 1.0971e-03 | 9.6872e-05 | ||||
20 | 2.1870e-02 | 1.9049 | 1.3910e-03 | 2.9512 | 6.7331e-05 | 4.0263 | 3.2018e-06 | 4.9191 |
30 | 1.0250e-02 | 1.8691 | 4.1775e-04 | 2.9667 | 1.3681e-05 | 3.9303 | 4.2892e-07 | 4.9578 |
40 | 5.9921e-03 | 1.8661 | 1.7881e-04 | 2.9496 | 4.4205e-06 | 3.9271 | 1.0318e-07 | 4.9527 |
50 | 3.9450e-03 | 1.8732 | 9.2744e-05 | 2.9420 | 1.8385e-06 | 3.9316 | 3.4162e-08 | 4.9536 |
60 | 2.7990e-03 | 1.8823 | 5.4251e-05 | 2.9411 | 8.9679e-07 | 3.9375 | 1.3837e-08 | 4.9570 |
70 | 2.0912e-03 | 1.8912 | 3.4467e-05 | 2.9427 | 4.8834e-07 | 3.9430 | 6.4409e-09 | 4.9606 |
80 | 1.6227e-03 | 1.8995 | 2.3258e-05 | 2.9458 | 2.8825e-07 | 3.9480 | 3.3196e-09 | 4.9638 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 8.1901e-02 | 1.0758e-02 | 1.0971e-03 | 9.6872e-05 | ||||
20 | 2.1870e-02 | 1.9049 | 1.3910e-03 | 2.9512 | 6.7331e-05 | 4.0263 | 3.2018e-06 | 4.9191 |
30 | 1.0250e-02 | 1.8691 | 4.1775e-04 | 2.9667 | 1.3681e-05 | 3.9303 | 4.2892e-07 | 4.9578 |
40 | 5.9921e-03 | 1.8661 | 1.7881e-04 | 2.9496 | 4.4205e-06 | 3.9271 | 1.0318e-07 | 4.9527 |
50 | 3.9450e-03 | 1.8732 | 9.2744e-05 | 2.9420 | 1.8385e-06 | 3.9316 | 3.4162e-08 | 4.9536 |
60 | 2.7990e-03 | 1.8823 | 5.4251e-05 | 2.9411 | 8.9679e-07 | 3.9375 | 1.3837e-08 | 4.9570 |
70 | 2.0912e-03 | 1.8912 | 3.4467e-05 | 2.9427 | 4.8834e-07 | 3.9430 | 6.4409e-09 | 4.9606 |
80 | 1.6227e-03 | 1.8995 | 2.3258e-05 | 2.9458 | 2.8825e-07 | 3.9480 | 3.3196e-09 | 4.9638 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 4.3648e-02 | 1.7020e-03 | 4.0015e-06 | 4.6584e-08 | ||||
20 | 5.6286e-03 | 2.9551 | 5.4668e-05 | 4.9604 | 6.6219e-08 | 5.9172 | 7.1450e-10 | 6.0268 |
30 | 1.6836e-03 | 2.9766 | 7.3187e-06 | 4.9594 | 8.1013e-09 | 5.1816 | 6.2456e-11 | 6.0107 |
40 | 7.1340e-04 | 2.9847 | 1.7583e-06 | 4.9572 | 1.9000e-09 | 5.0409 | 1.1093e-11 | 6.0071 |
50 | 3.6619e-04 | 2.9886 | 5.8254e-07 | 4.9506 | 6.2116e-10 | 5.0103 | 2.9045e-12 | 6.0053 |
60 | 2.1226e-04 | 2.9911 | 2.3667e-07 | 4.9403 | 2.4956e-10 | 5.0015 | 9.7196e-13 | 6.0042 |
70 | 1.3382e-04 | 2.9926 | 1.1074e-07 | 4.9269 | 1.1549e-10 | 4.9985 | 3.9620e-13 | 5.8215 |
80 | 8.9728e-05 | 2.9934 | 5.7480e-08 | 4.9108 | 5.9257e-11 | 4.9973 | 1.8005e-13 | 5.9064 |
p=1 | p=2 | p=3 | p=4 | |||||
N | order | order | order | order | ||||
10 | 4.3648e-02 | 1.7020e-03 | 4.0015e-06 | 4.6584e-08 | ||||
20 | 5.6286e-03 | 2.9551 | 5.4668e-05 | 4.9604 | 6.6219e-08 | 5.9172 | 7.1450e-10 | 6.0268 |
30 | 1.6836e-03 | 2.9766 | 7.3187e-06 | 4.9594 | 8.1013e-09 | 5.1816 | 6.2456e-11 | 6.0107 |
40 | 7.1340e-04 | 2.9847 | 1.7583e-06 | 4.9572 | 1.9000e-09 | 5.0409 | 1.1093e-11 | 6.0071 |
50 | 3.6619e-04 | 2.9886 | 5.8254e-07 | 4.9506 | 6.2116e-10 | 5.0103 | 2.9045e-12 | 6.0053 |
60 | 2.1226e-04 | 2.9911 | 2.3667e-07 | 4.9403 | 2.4956e-10 | 5.0015 | 9.7196e-13 | 6.0042 |
70 | 1.3382e-04 | 2.9926 | 1.1074e-07 | 4.9269 | 1.1549e-10 | 4.9985 | 3.9620e-13 | 5.8215 |
80 | 8.9728e-05 | 2.9934 | 5.7480e-08 | 4.9108 | 5.9257e-11 | 4.9973 | 1.8005e-13 | 5.9064 |
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