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Combinatorics of some fifth and sixth order mock theta functions
A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations
1. | School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China |
2. | School of Mathematical Sciences, Xiamen University, Fujian Provincial Key Laboratory of Mathematical Modeling, and High-Performance Scientific Computing, Xiamen, Fujian 361005, China |
In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [
References:
[1] |
S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation, submitted (2019), arXiv: 1905.03660. Google Scholar |
[2] |
X. Cai, S. Boscarino and J.-M. Qiu, High order semi-Lagrangian discontinuous galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics, submitted (2019), arXiv: 1911.12229. Google Scholar |
[3] |
X. Cai, W. Guo and J.-M. Qiu,
A high order conservative semi-Lagrangian discontinuous Galerkin method for two-dimensional transport simulations, Journal of Scientific Computing, 73 (2017), 514-542.
doi: 10.1007/s10915-017-0554-0. |
[4] |
X. Cai, J.-X. Qiu and J.-M. Qiu,
A conservative semi-Lagrangian HWENO method for the Vlasov equation, Journal of Computational Physics, 323 (2016), 95-114.
doi: 10.1016/j.jcp.2016.07.021. |
[5] |
E. Celledoni and B. K. Kometa,
Semi-Lagrangian Runge-Kutta exponential integrators for convection dominated problems, Journal of Scientific Computing, 41 (2009), 139-164.
doi: 10.1007/s10915-009-9291-3. |
[6] |
E. Celledoni, B. K. Kometa and O. Verdier,
High order semi-Lagrangian methods for the incompressible Navier–Stokes equations, Journal of Scientific Computing, 66 (2016), 91-115.
doi: 10.1007/s10915-015-0015-6. |
[7] |
E. Celledoni, A. Marthinsen and B. Owren,
Commutator-free Lie group methods, Future Generation Computer Systems, 19 (2003), 341-352.
doi: 10.1016/S0167-739X(02)00161-9. |
[8] |
G.-H. Cottet, J.-M. Etancelin, F. Perignon and C. Picard,
High order semi-Lagrangian particles for transport equations: Numerical analysis and implementation issues, ESIAM: Mathematical Modelling and Numerical Analysis, 48 (2014), 1029-1060.
doi: 10.1051/m2an/2014009. |
[9] |
N. Crouseilles, M. Mehrenberger and E. Sonnendrücker,
Conservative semi-Lagrangian schemes for Vlasov equations, Journal of Computational Physics, 229 (2010), 1927-1953.
doi: 10.1016/j.jcp.2009.11.007. |
[10] |
N. Crouseilles, T. Respaud and E. Sonnendrücke,
A forward semi-Lagrangian method for the numerical solution of the Vlasov equation, Computer Physics Communications, 180 (2009), 1730-1745.
doi: 10.1016/j.cpc.2009.04.024. |
[11] |
K. Duraisamy and J. D. Baeder,
Implicit scheme for hyperbolic conservation laws using nonoscillatory reconstruction in space and time, SIAM Journal on Scientific Computing, 29 (2007), 2607-2620.
doi: 10.1137/070683271. |
[12] |
A. Efremov, E. Karepova and V. Shaydurov, A conservative semi-Lagrangian method for the advection problem, Numerical Analysis and its Applications, Lecture Notes in Comput. Sci., Springer, Cham, 10187 (2017), 325–333.
doi: 10.1007/978-3-319-57099-0. |
[13] |
L. Fatone, D. Funaro and G. Manzini,
A semi-Lagrangian spectral method for the Vlasov-Poisson system based on Fourier, Legendre and Hermite polynomials, Communications on Applied Mathematics and Computation, 1 (2019), 333-360.
doi: 10.1007/s42967-019-00027-8. |
[14] |
F. Filbet and C. Prouveur,
High order time discretization for backward semi-Lagrangian methods, Journal of Computational and Applied Mathematics, 303 (2016), 171-188.
doi: 10.1016/j.cam.2016.01.024. |
[15] |
W. Guo, R. D. Nair and J.-M. Qiu,
A conservative semi-Lagrangian discontinuous Galerkin scheme on the cubed-sphere, Monthly Weather Review, 142 (2014), 457-475.
doi: 10.1175/MWR-D-13-00048.1. |
[16] |
C.-S. Huang, T. Arbogast and C.-H. Hung,
A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws, Journal of Computational Physics, 322 (2016), 559-585.
doi: 10.1016/j.jcp.2016.06.027. |
[17] |
R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[18] |
J.-M. Qiu,
High order mass conservative semi-Lagrangian methods for transport problems, Handbook of Numerical Methods for Hyperbolic Problems, 17 (2016), 353-382.
|
[19] |
J.-M. Qiu and A. Christlieb,
A conservative high order semi-Lagrangian WENO method for the Vlasov equation, Journal of Computational Physics, 229 (2010), 1130-1149.
doi: 10.1016/j.jcp.2009.10.016. |
[20] |
J.-M. Qiu and G. Russo,
A high order multi-dimensional characteristic tracing strategy for the Vlasov-Poisson System, Journal of Scientific Computing, 71 (2017), 414-434.
doi: 10.1007/s10915-016-0305-7. |
[21] |
J.-M. Qiu and C.-W. Shu,
Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow, Journal of Computational Physics, 230 (2011), 863-889.
doi: 10.1016/j.jcp.2010.04.037. |
[22] |
J.-M. Qiu and C.-W. Shu,
Convergence of Godunov-type schemes for scalar conservation laws under large time steps, SIAM Journal on Numerical Analysis, 46 (2008), 2211-2237.
doi: 10.1137/060657911. |
[23] |
J. A. Rossmanith and D. C. Seal,
A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations, Journal of Computational Physics, 230 (2011), 6203-6232.
doi: 10.1016/j.jcp.2011.04.018. |
[24] |
C.-W. Shu,
High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Review, 51 (2009), 82-126.
doi: 10.1137/070679065. |
[25] |
D. Sirajuddin and W. N. G. Hitchon,
A truly forward semi-Lagrangian WENO scheme for the Vlasov-Poisson system, Journal of Computational Physics, 392 (2019), 619-665.
doi: 10.1016/j.jcp.2019.04.054. |
[26] |
E. Sonnendruüker, J. Roche, P. Bertrand and A. Ghizzo,
The semi-Lagrangian method for the numerical resolution of the Vlasov equation, Journal of Computational Physics, 149 (1999), 201-220.
doi: 10.1006/jcph.1998.6148. |
[27] |
A. Staniforth and J. Cȏté,
Semi-Lagrangian integration schemes for atmospheric models: A review, Monthly Weather Review, 119 (1991), 2206-2223.
doi: 10.1175/1520-0493(1991)119<2206:SLISFA>2.0.CO;2. |
[28] |
G. Tumolo, L. Bonaventura and M. Restelli,
A semi-implicit, semi-Lagrangian, $p$-adaptive discontinuous Galerkin method for the shallow water equations, Journal of Computational Physics, 232 (2013), 46-67.
doi: 10.1016/j.jcp.2012.06.006. |
[29] |
T. Xiong, J.-M. Qiu, Z. Xu and A. Christlieb,
High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation, Journal of Computational Physics, 273 (2014), 618-639.
doi: 10.1016/j.jcp.2014.05.033. |
[30] |
T. Xiong, G. Russo and J.-M. Qiu,
High order multi-dimensional characteristics tracing for the incompressible Euler equation and the guiding-center Vlasov equation, Journal of Scientific Computing, 77 (2018), 263-282.
doi: 10.1007/s10915-018-0705-y. |
[31] |
T. Xiong, G. Russo and J.-M. Qiu,
Conservative multi-dimensional semi-Lagrangian finite difference scheme: Stability and applications to the kinetic and fluid simulations, Journal of Scientific Computing, 79 (2019), 1241-1270.
doi: 10.1007/s10915-018-0892-6. |
[32] |
D. Xiu and G. E. Karniadakis,
A semi-Lagrangian high-order method for Navier-Stokes equations, Journal of Computational Physics, 172 (2001), 658-684.
doi: 10.1006/jcph.2001.6847. |
[33] |
T. Yabe and Y. Ogata,
Conservative semi-Lagrangian CIP technique for the shallow water equations, Computational Mechanics, 46 (2010), 125-134.
doi: 10.1007/s00466-009-0438-8. |
show all references
References:
[1] |
S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation, submitted (2019), arXiv: 1905.03660. Google Scholar |
[2] |
X. Cai, S. Boscarino and J.-M. Qiu, High order semi-Lagrangian discontinuous galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics, submitted (2019), arXiv: 1911.12229. Google Scholar |
[3] |
X. Cai, W. Guo and J.-M. Qiu,
A high order conservative semi-Lagrangian discontinuous Galerkin method for two-dimensional transport simulations, Journal of Scientific Computing, 73 (2017), 514-542.
doi: 10.1007/s10915-017-0554-0. |
[4] |
X. Cai, J.-X. Qiu and J.-M. Qiu,
A conservative semi-Lagrangian HWENO method for the Vlasov equation, Journal of Computational Physics, 323 (2016), 95-114.
doi: 10.1016/j.jcp.2016.07.021. |
[5] |
E. Celledoni and B. K. Kometa,
Semi-Lagrangian Runge-Kutta exponential integrators for convection dominated problems, Journal of Scientific Computing, 41 (2009), 139-164.
doi: 10.1007/s10915-009-9291-3. |
[6] |
E. Celledoni, B. K. Kometa and O. Verdier,
High order semi-Lagrangian methods for the incompressible Navier–Stokes equations, Journal of Scientific Computing, 66 (2016), 91-115.
doi: 10.1007/s10915-015-0015-6. |
[7] |
E. Celledoni, A. Marthinsen and B. Owren,
Commutator-free Lie group methods, Future Generation Computer Systems, 19 (2003), 341-352.
doi: 10.1016/S0167-739X(02)00161-9. |
[8] |
G.-H. Cottet, J.-M. Etancelin, F. Perignon and C. Picard,
High order semi-Lagrangian particles for transport equations: Numerical analysis and implementation issues, ESIAM: Mathematical Modelling and Numerical Analysis, 48 (2014), 1029-1060.
doi: 10.1051/m2an/2014009. |
[9] |
N. Crouseilles, M. Mehrenberger and E. Sonnendrücker,
Conservative semi-Lagrangian schemes for Vlasov equations, Journal of Computational Physics, 229 (2010), 1927-1953.
doi: 10.1016/j.jcp.2009.11.007. |
[10] |
N. Crouseilles, T. Respaud and E. Sonnendrücke,
A forward semi-Lagrangian method for the numerical solution of the Vlasov equation, Computer Physics Communications, 180 (2009), 1730-1745.
doi: 10.1016/j.cpc.2009.04.024. |
[11] |
K. Duraisamy and J. D. Baeder,
Implicit scheme for hyperbolic conservation laws using nonoscillatory reconstruction in space and time, SIAM Journal on Scientific Computing, 29 (2007), 2607-2620.
doi: 10.1137/070683271. |
[12] |
A. Efremov, E. Karepova and V. Shaydurov, A conservative semi-Lagrangian method for the advection problem, Numerical Analysis and its Applications, Lecture Notes in Comput. Sci., Springer, Cham, 10187 (2017), 325–333.
doi: 10.1007/978-3-319-57099-0. |
[13] |
L. Fatone, D. Funaro and G. Manzini,
A semi-Lagrangian spectral method for the Vlasov-Poisson system based on Fourier, Legendre and Hermite polynomials, Communications on Applied Mathematics and Computation, 1 (2019), 333-360.
doi: 10.1007/s42967-019-00027-8. |
[14] |
F. Filbet and C. Prouveur,
High order time discretization for backward semi-Lagrangian methods, Journal of Computational and Applied Mathematics, 303 (2016), 171-188.
doi: 10.1016/j.cam.2016.01.024. |
[15] |
W. Guo, R. D. Nair and J.-M. Qiu,
A conservative semi-Lagrangian discontinuous Galerkin scheme on the cubed-sphere, Monthly Weather Review, 142 (2014), 457-475.
doi: 10.1175/MWR-D-13-00048.1. |
[16] |
C.-S. Huang, T. Arbogast and C.-H. Hung,
A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws, Journal of Computational Physics, 322 (2016), 559-585.
doi: 10.1016/j.jcp.2016.06.027. |
[17] |
R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0348-8629-1. |
[18] |
J.-M. Qiu,
High order mass conservative semi-Lagrangian methods for transport problems, Handbook of Numerical Methods for Hyperbolic Problems, 17 (2016), 353-382.
|
[19] |
J.-M. Qiu and A. Christlieb,
A conservative high order semi-Lagrangian WENO method for the Vlasov equation, Journal of Computational Physics, 229 (2010), 1130-1149.
doi: 10.1016/j.jcp.2009.10.016. |
[20] |
J.-M. Qiu and G. Russo,
A high order multi-dimensional characteristic tracing strategy for the Vlasov-Poisson System, Journal of Scientific Computing, 71 (2017), 414-434.
doi: 10.1007/s10915-016-0305-7. |
[21] |
J.-M. Qiu and C.-W. Shu,
Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow, Journal of Computational Physics, 230 (2011), 863-889.
doi: 10.1016/j.jcp.2010.04.037. |
[22] |
J.-M. Qiu and C.-W. Shu,
Convergence of Godunov-type schemes for scalar conservation laws under large time steps, SIAM Journal on Numerical Analysis, 46 (2008), 2211-2237.
doi: 10.1137/060657911. |
[23] |
J. A. Rossmanith and D. C. Seal,
A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations, Journal of Computational Physics, 230 (2011), 6203-6232.
doi: 10.1016/j.jcp.2011.04.018. |
[24] |
C.-W. Shu,
High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Review, 51 (2009), 82-126.
doi: 10.1137/070679065. |
[25] |
D. Sirajuddin and W. N. G. Hitchon,
A truly forward semi-Lagrangian WENO scheme for the Vlasov-Poisson system, Journal of Computational Physics, 392 (2019), 619-665.
doi: 10.1016/j.jcp.2019.04.054. |
[26] |
E. Sonnendruüker, J. Roche, P. Bertrand and A. Ghizzo,
The semi-Lagrangian method for the numerical resolution of the Vlasov equation, Journal of Computational Physics, 149 (1999), 201-220.
doi: 10.1006/jcph.1998.6148. |
[27] |
A. Staniforth and J. Cȏté,
Semi-Lagrangian integration schemes for atmospheric models: A review, Monthly Weather Review, 119 (1991), 2206-2223.
doi: 10.1175/1520-0493(1991)119<2206:SLISFA>2.0.CO;2. |
[28] |
G. Tumolo, L. Bonaventura and M. Restelli,
A semi-implicit, semi-Lagrangian, $p$-adaptive discontinuous Galerkin method for the shallow water equations, Journal of Computational Physics, 232 (2013), 46-67.
doi: 10.1016/j.jcp.2012.06.006. |
[29] |
T. Xiong, J.-M. Qiu, Z. Xu and A. Christlieb,
High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation, Journal of Computational Physics, 273 (2014), 618-639.
doi: 10.1016/j.jcp.2014.05.033. |
[30] |
T. Xiong, G. Russo and J.-M. Qiu,
High order multi-dimensional characteristics tracing for the incompressible Euler equation and the guiding-center Vlasov equation, Journal of Scientific Computing, 77 (2018), 263-282.
doi: 10.1007/s10915-018-0705-y. |
[31] |
T. Xiong, G. Russo and J.-M. Qiu,
Conservative multi-dimensional semi-Lagrangian finite difference scheme: Stability and applications to the kinetic and fluid simulations, Journal of Scientific Computing, 79 (2019), 1241-1270.
doi: 10.1007/s10915-018-0892-6. |
[32] |
D. Xiu and G. E. Karniadakis,
A semi-Lagrangian high-order method for Navier-Stokes equations, Journal of Computational Physics, 172 (2001), 658-684.
doi: 10.1006/jcph.2001.6847. |
[33] |
T. Yabe and Y. Ogata,
Conservative semi-Lagrangian CIP technique for the shallow water equations, Computational Mechanics, 46 (2010), 125-134.
doi: 10.1007/s00466-009-0438-8. |




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Order | Order | |||
20 | 1.25E-4 | – | 2.07E-4 | – |
40 | 3.83E-6 | 5.03 | 7.8E-6 | 4.73 |
80 | 1.15E-7 | 5.06 | 2.38E-7 | 5.04 |
160 | 3.55E-9 | 5.00 | 7.29E-9 | 5.03 |
320 | 1.10E-10 | 5.01 | 2.00E-10 | 5.19 |
640 | 3.42E-12 | 5.01 | 6.03E-12 | 5.05 |
Order | Order | |||
20 | 1.25E-4 | – | 2.07E-4 | – |
40 | 3.83E-6 | 5.03 | 7.8E-6 | 4.73 |
80 | 1.15E-7 | 5.06 | 2.38E-7 | 5.04 |
160 | 3.55E-9 | 5.00 | 7.29E-9 | 5.03 |
320 | 1.10E-10 | 5.01 | 2.00E-10 | 5.19 |
640 | 3.42E-12 | 5.01 | 6.03E-12 | 5.05 |
3rd order Scheme | 4th order scheme | |||||||
Order | Order | Order | Order | |||||
80 | 6.98E-4 | – | 4.25E-3 | – | 9.99E-5 | – | 1.30E-3 | – |
160 | 9.11E-5 | 2.94 | 6.15E-4 | 2.79 | 6.07E-6 | 4.04 | 2.00E-4 | 2.71 |
320 | 1.17E-5 | 2.96 | 7.92E-5 | 2.96 | 2.81E-7 | 4.43 | 1.62E-5 | 3.62 |
640 | 1.53E-6 | 2.94 | 1.08E-5 | 2.88 | 2.28E-8 | 3.63 | 1.27E-6 | 3.67 |
1280 | 1.95E-7 | 2.97 | 1.69E-6 | 2.95 | 3.09E-9 | 2.88 | 6.38E-8 | 4.32 |
2560 | 2.47E-8 | 2.98 | 1.77E-7 | 2.98 | 4.46E-10 | 2.79 | 7.59E-9 | 3.07 |
3rd order Scheme | 4th order scheme | |||||||
Order | Order | Order | Order | |||||
80 | 6.98E-4 | – | 4.25E-3 | – | 9.99E-5 | – | 1.30E-3 | – |
160 | 9.11E-5 | 2.94 | 6.15E-4 | 2.79 | 6.07E-6 | 4.04 | 2.00E-4 | 2.71 |
320 | 1.17E-5 | 2.96 | 7.92E-5 | 2.96 | 2.81E-7 | 4.43 | 1.62E-5 | 3.62 |
640 | 1.53E-6 | 2.94 | 1.08E-5 | 2.88 | 2.28E-8 | 3.63 | 1.27E-6 | 3.67 |
1280 | 1.95E-7 | 2.97 | 1.69E-6 | 2.95 | 3.09E-9 | 2.88 | 6.38E-8 | 4.32 |
2560 | 2.47E-8 | 2.98 | 1.77E-7 | 2.98 | 4.46E-10 | 2.79 | 7.59E-9 | 3.07 |
3rd order RKEI (2) | 4th order RKEI (3) | |||||||
Order | Order | Order | Order | |||||
40 | 1.54E-5 | – | 2.39E-5 | – | 5.95E-8 | – | 8.80E-8 | – |
39 | 1.39E-5 | 3.02 | 2.19E-5 | 2.58 | 5.13E-8 | 4.37 | 7.78E-8 | 3.64 |
38 | 1.25E-5 | 3.03 | 1.99E-5 | 2.73 | 4.41E-8 | 4.31 | 6.85E-8 | 3.63 |
37 | 1.12E-5 | 3.02 | 1.74E-5 | 3.69 | 3.88E-8 | 3.52 | 5.76E-8 | 4.77 |
36 | 1.00E-5 | 3.00 | 1.57E-5 | 2.73 | 3.29E-8 | 4.37 | 5.03E-8 | 3.59 |
3rd order RKEI (2) | 4th order RKEI (3) | |||||||
Order | Order | Order | Order | |||||
40 | 1.54E-5 | – | 2.39E-5 | – | 5.95E-8 | – | 8.80E-8 | – |
39 | 1.39E-5 | 3.02 | 2.19E-5 | 2.58 | 5.13E-8 | 4.37 | 7.78E-8 | 3.64 |
38 | 1.25E-5 | 3.03 | 1.99E-5 | 2.73 | 4.41E-8 | 4.31 | 6.85E-8 | 3.63 |
37 | 1.12E-5 | 3.02 | 1.74E-5 | 3.69 | 3.88E-8 | 3.52 | 5.76E-8 | 4.77 |
36 | 1.00E-5 | 3.00 | 1.57E-5 | 2.73 | 3.29E-8 | 4.37 | 5.03E-8 | 3.59 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 6.40E-5 | – | 1.36E-4 | – | 2.24E-4 | – |
80 | 7.42E-6 | 3.11 | 1.03E-5 | 3.72 | 2.30E-5 | 3.28 |
160 | 9.57E-7 | 2.95 | 9.29E-7 | 3.47 | 2.58E-6 | 3.16 |
320 | 1.35E-7 | 2.83 | 1.04E-7 | 3.16 | 3.24E-7 | 2.99 |
640 | 1.73E-8 | 2.97 | 1.17E-8 | 3.15 | 3.93E-8 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 6.40E-5 | – | 1.36E-4 | – | 2.24E-4 | – |
80 | 7.42E-6 | 3.11 | 1.03E-5 | 3.72 | 2.30E-5 | 3.28 |
160 | 9.57E-7 | 2.95 | 9.29E-7 | 3.47 | 2.58E-6 | 3.16 |
320 | 1.35E-7 | 2.83 | 1.04E-7 | 3.16 | 3.24E-7 | 2.99 |
640 | 1.73E-8 | 2.97 | 1.17E-8 | 3.15 | 3.93E-8 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 3.35E-4 | – | 9.72E-4 | – | 1.20E-3 | – |
80 | 4.75E-5 | 2.82 | 7.51E-5 | 3.69 | 1.30E-4 | 3.26 |
160 | 6.17E-6 | 2.95 | 6.16E-6 | 3.61 | 1.59E-5 | 3.03 |
320 | 8.72E-7 | 2.82 | 7.09E-7 | 3.12 | 2.19E-6 | 2.86 |
640 | 1.10E-7 | 2.98 | 7.64E-8 | 3.22 | 2.66E-7 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 3.35E-4 | – | 9.72E-4 | – | 1.20E-3 | – |
80 | 4.75E-5 | 2.82 | 7.51E-5 | 3.69 | 1.30E-4 | 3.26 |
160 | 6.17E-6 | 2.95 | 6.16E-6 | 3.61 | 1.59E-5 | 3.03 |
320 | 8.72E-7 | 2.82 | 7.09E-7 | 3.12 | 2.19E-6 | 2.86 |
640 | 1.10E-7 | 2.98 | 7.64E-8 | 3.22 | 2.66E-7 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.19E-6 | – | 1.12E-4 | – | 1.12E-4 | – |
80 | 2.41E-7 | 4.43 | 7.04E-6 | 3.99 | 6.64E-6 | 4.08 |
160 | 1.09E-8 | 4.47 | 4.67E-7 | 3.91 | 3.95E-7 | 4.07 |
320 | 5.96E-10 | 4.19 | 4.16E-8 | 3.49 | 2.97E-8 | 3.74 |
640 | 3.23E-11 | 4.21 | 4.06E-9 | 3.36 | 2.29E-9 | 3.69 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.19E-6 | – | 1.12E-4 | – | 1.12E-4 | – |
80 | 2.41E-7 | 4.43 | 7.04E-6 | 3.99 | 6.64E-6 | 4.08 |
160 | 1.09E-8 | 4.47 | 4.67E-7 | 3.91 | 3.95E-7 | 4.07 |
320 | 5.96E-10 | 4.19 | 4.16E-8 | 3.49 | 2.97E-8 | 3.74 |
640 | 3.23E-11 | 4.21 | 4.06E-9 | 3.36 | 2.29E-9 | 3.69 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.66E-5 | – | 9.66E-4 | – | 1.00E-3 | – |
80 | 2.80E-6 | 4.34 | 6.27E-5 | 3.95 | 6.33E-5 | 3.99 |
160 | 8.62E-8 | 5.02 | 3.94E-6 | 3.99 | 3.82E-6 | 4.05 |
320 | 4.01E-9 | 4.42 | 3.43E-7 | 3.52 | 3.16E-7 | 3.59 |
640 | 2.82E-10 | 3.83 | 2.79E-8 | 3.62 | 2.38E-8 | 3.73 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.66E-5 | – | 9.66E-4 | – | 1.00E-3 | – |
80 | 2.80E-6 | 4.34 | 6.27E-5 | 3.95 | 6.33E-5 | 3.99 |
160 | 8.62E-8 | 5.02 | 3.94E-6 | 3.99 | 3.82E-6 | 4.05 |
320 | 4.01E-9 | 4.42 | 3.43E-7 | 3.52 | 3.16E-7 | 3.59 |
640 | 2.82E-10 | 3.83 | 2.79E-8 | 3.62 | 2.38E-8 | 3.73 |
CFL | 3rd order SLFDn | 4th order SLFDn | ||||||
Order | Order | Order | Order | |||||
3 | 4.51E-8 | – | 9.66E-8 | – | 2.33E-10 | – | 2.01E-9 | – |
2.9 | 4.07E-8 | 3.03 | 8.72E-8 | 3.02 | 2.03E-10 | 4.07 | 1.74E-9 | 4.23 |
2.8 | 3.72E-8 | 2.56 | 7.93E-8 | 2.71 | 1.88E-10 | 2.19 | 1.65E-9 | 1.52 |
2.7 | 3.32E-8 | 3.13 | 7.09E-8 | 3.08 | 1.59E-10 | 4.61 | 1.39E-9 | 4.54 |
2.6 | 2.94E-8 | 3.22 | 6.30E-8 | 3.13 | 1.33E-10 | 4.73 | 1.15E-9 | 5.02 |
CFL | 3rd order SLFDn | 4th order SLFDn | ||||||
Order | Order | Order | Order | |||||
3 | 4.51E-8 | – | 9.66E-8 | – | 2.33E-10 | – | 2.01E-9 | – |
2.9 | 4.07E-8 | 3.03 | 8.72E-8 | 3.02 | 2.03E-10 | 4.07 | 1.74E-9 | 4.23 |
2.8 | 3.72E-8 | 2.56 | 7.93E-8 | 2.71 | 1.88E-10 | 2.19 | 1.65E-9 | 1.52 |
2.7 | 3.32E-8 | 3.13 | 7.09E-8 | 3.08 | 1.59E-10 | 4.61 | 1.39E-9 | 4.54 |
2.6 | 2.94E-8 | 3.22 | 6.30E-8 | 3.13 | 1.33E-10 | 4.73 | 1.15E-9 | 5.02 |
CFL | 3rd order SLFDc2 | 3rd order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
2 | 4.67E-8 | – | 3.20E-7 | – | 3.07E-8 | – | 1.28E-7 | – |
1.9 | 4.08E-8 | 2.63 | 2.84E-7 | 2.33 | 2.78E-8 | 1.93 | 1.16E-7 | 1.92 |
1.8 | 3.50E-8 | 2.84 | 2.45E-7 | 2.73 | 2.45E-8 | 2.33 | 9.83E-8 | 3.06 |
1.7 | 2.95E-8 | 2.99 | 2.05E-7 | 3.12 | 2.21E-8 | 1.81 | 8.98E-8 | 1.58 |
1.6 | 2.44E-8 | 3.13 | 1.67E-7 | 3.38 | 1.99E-8 | 1.73 | 8.21E-8 | 1.48 |
CFL | 3rd order SLFDc2 | 3rd order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
2 | 4.67E-8 | – | 3.20E-7 | – | 3.07E-8 | – | 1.28E-7 | – |
1.9 | 4.08E-8 | 2.63 | 2.84E-7 | 2.33 | 2.78E-8 | 1.93 | 1.16E-7 | 1.92 |
1.8 | 3.50E-8 | 2.84 | 2.45E-7 | 2.73 | 2.45E-8 | 2.33 | 9.83E-8 | 3.06 |
1.7 | 2.95E-8 | 2.99 | 2.05E-7 | 3.12 | 2.21E-8 | 1.81 | 8.98E-8 | 1.58 |
1.6 | 2.44E-8 | 3.13 | 1.67E-7 | 3.38 | 1.99E-8 | 1.73 | 8.21E-8 | 1.48 |
CFL | 4th order SLFDc2 | 4th order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
3 | 5.46E-10 | – | 2.15E-9 | – | 2.53E-10 | – | 7.94E-10 | – |
2.9 | 4.75E-10 | 4.11 | 1.70E-9 | 6.93 | 2.37E-10 | 1.92 | 6.58E-10 | 5.54 |
2.8 | 4.16E-10 | 3.78 | 1.52E-9 | 3.19 | 2.32E-10 | 0.61 | 5.65E-10 | 4.34 |
2.7 | 3.66E-10 | 3.52 | 1.29E-9 | 4.51 | 2.23E-10 | 1.09 | 4.51E-10 | 6.20 |
2.6 | 3.25E-10 | 3.15 | 1.08E-9 | 4.71 | 2.15E-10 | 0.97 | 3.44E-10 | 7.17 |
CFL | 4th order SLFDc2 | 4th order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
3 | 5.46E-10 | – | 2.15E-9 | – | 2.53E-10 | – | 7.94E-10 | – |
2.9 | 4.75E-10 | 4.11 | 1.70E-9 | 6.93 | 2.37E-10 | 1.92 | 6.58E-10 | 5.54 |
2.8 | 4.16E-10 | 3.78 | 1.52E-9 | 3.19 | 2.32E-10 | 0.61 | 5.65E-10 | 4.34 |
2.7 | 3.66E-10 | 3.52 | 1.29E-9 | 4.51 | 2.23E-10 | 1.09 | 4.51E-10 | 6.20 |
2.6 | 3.25E-10 | 3.15 | 1.08E-9 | 4.71 | 2.15E-10 | 0.97 | 3.44E-10 | 7.17 |
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