American Institute of Mathematical Sciences

Advanced
doi: 10.3934/era.2020093

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

Guoliang Zhang 1, , Shaoqin Zheng 1, and  Tao Xiong 2,,

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

* Corresponding author: Tao Xiong

Special issue "High-order numerical methods for PDEs & applications"

Received  March 2020 Revised  July 2020 Published  September 2020

Fund Project: The third author is supported by NSF grant of Fujian Province No. 2019J06002, NSFC Grant No. 11971025, Science Challenge Project No. TZ2016002, NSAF Grant No. U1630247

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 [21], with a high order RKEI method [7], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.

Keywords: Semi-Lagrangian, finite difference WENO scheme, scalar nonlinear hyperbolic equations, Runge-Kutta exponential integrator, conservation.
Mathematics Subject Classification: 65M06, 65M25.
Citation: Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, doi: 10.3934/era.2020093
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Review, 51 (2009), 82-126.  doi: 10.1137/070679065.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Review, 51 (2009), 82-126.  doi: 10.1137/070679065.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The diagram of the spatio-temporal region $ \Omega_{i,t} $
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Figure 1.  Numerical errors as compared to the exact solution for three schemes at $ T = 0.5/\pi $ for the nonlinear Burgers' equation (44). The solid line is nonconservative scheme SLFDn; the symbol "o" is the conservative scheme SLFDc1 and the symbol "*" is the conservative scheme SLFDc2. $ \text{CFL} = 1.5 $ and $ N = 320 $. Left: 3rd order RKEI (2); right: 4th order RKEI (3)
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Figure 2.  The shock wave solution at $ T = 2/\pi $ for the nonlinear Burgers' equation (44). The solid line is the exact solution. The dashed line is the 2nd order nonconservative scheme SLFDn; the symbol "o" is the conservative scheme SLFDc1 and the symbol $ \times $ is the conservative scheme SLFDc2. $ \text{CFL} = 1.5 $. Left: 3rd order RKEI (2); right: 4th order RKEI (3)
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Figure 3.  Numerical solutions for the nonlinear Burgers' equation (44) with initial condition $ \sqrt{2}/2+\sin(\pi x) $. The solid line is the exact solution. The dashed line is the 2nd order nonconservative scheme SLFDn; the symbol "o" is the conservative scheme SLFDc1 and the symbol "+" is the conservative scheme SLFDc2. $ \text{CFL} = 1.5 $ and $ N = 88 $. 3rd order RKEI is used. Top left: $ T = 0.7/\pi $; top right: $ T = 1.3/\pi $; bottom: $ T = 2/\pi $
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Table 1.  A Butcher tableau for RKEI method, where $ a_{ik} = \sum_{l = 1}^{J^{(i)}} \alpha_{i,l}^k $ and $ b_k = \sum_{l = 1}^{J} \beta_l^k $, which merges $ J^{(i)} $ rows into one row at each stage $ i $
$ \mathbf{c} $ $ A $
$ \mathbf{b} $
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$ \mathbf{c} $ $ A $
$ \mathbf{b} $
Table 2.  CF3
$ 0 $
$ \frac12 $ $ \frac12 $
$ 1 $ $ -1 $ $ 2 $
$ \frac{1}{12} $ $ \frac13 $ -$ \frac14 $
$ \frac{1}{12} $ $ \frac13 $ $ \frac{5}{12} $
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$ 0 $
$ \frac12 $ $ \frac12 $
$ 1 $ $ -1 $ $ 2 $
$ \frac{1}{12} $ $ \frac13 $ -$ \frac14 $
$ \frac{1}{12} $ $ \frac13 $ $ \frac{5}{12} $
Table 3.  CF4
$ 0 $
$ \frac12 $ $ \frac12 $
$ \frac12 $ $ 0 $ $ \frac12 $
$ 1 $ $ \frac12 $ $ 0 $ $ 0 $
$ -\frac12 $ $ 0 $ $ 1 $
$ \frac{1}{4} $ $ \frac16 $ $ \frac16 $ -$ \frac{1}{12} $
$ -\frac{1}{12} $ $ \frac16 $ $ \frac16 $ $ \frac{1}{4} $
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$ 0 $
$ \frac12 $ $ \frac12 $
$ \frac12 $ $ 0 $ $ \frac12 $
$ 1 $ $ \frac12 $ $ 0 $ $ 0 $
$ -\frac12 $ $ 0 $ $ 1 $
$ \frac{1}{4} $ $ \frac16 $ $ \frac16 $ -$ \frac{1}{12} $
$ -\frac{1}{12} $ $ \frac16 $ $ \frac16 $ $ \frac{1}{4} $
Table 1.  Numerical errors and orders for the linear problem (41). $ T = 2.5 $ and $ \text{CFL} = 4.5 $ in (40)
$ N $ $ L_1 $ error Order $ L_{\infty} $ error 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
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$ N $ $ L_1 $ error Order $ L_{\infty} $ error 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
Table 2.  Numerical errors and orders for the quasilinear case (42). $ T = 1.5 $ and $ \text{CFL} = 4.5 $ in (40)
3rd order Scheme 4th order scheme
$ N $ $ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
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3rd order Scheme 4th order scheme
$ N $ $ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
Table 3.  Numerical errors and orders for the quasilinear case (42) with different CFL numbers. $ N = 900 $ and $ T = 1.5 $
$ \text{CFL} $ 3rd order RKEI (2) 4th order RKEI (3)
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
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$ \text{CFL} $ 3rd order RKEI (2) 4th order RKEI (3)
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
Table 4.  Numerical $ L_1 $ errors and orders for the nonlinear Burgers' equation (44). $ T = 0.5/\pi $ and $ \text{CFL} = 1.5 $ in (40). 3rd order RKEI (2) is used
SLFDn SLFDc1 SLFDc2
$ N $ $ L_1 $ error Order $ L_1 $ error Order $ L_1 $ error 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
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SLFDn SLFDc1 SLFDc2
$ N $ $ L_1 $ error Order $ L_1 $ error Order $ L_1 $ error 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
Table 5.  Numerical $ L_\infty $ errors and orders for the nonlinear Burgers' equation (44). $ T = 0.5/\pi $ and $ \text{CFL} = 1.5 $ in (40). 3rd order RKEI (2) is used
SLFDn SLFDc1 SLFDc2
$ N $ $ L_{\infty} $ error Order $ L_{\infty} $ error Order $ L_{\infty} $ error 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
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SLFDn SLFDc1 SLFDc2
$ N $ $ L_{\infty} $ error Order $ L_{\infty} $ error Order $ L_{\infty} $ error 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
Table 6.  Numerical $ L_1 $ errors and orders for the nonlinear Burgers' equation (44). $ T = 0.5/\pi $ and $ \text{CFL} = 1.5 $ in (40). 4th order RKEI (3) is used
SLFDn SLFDc1 SLFDc2
$ N $ $ L_1 $ error Order $ L_1 $ error Order $ L_1 $ error 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
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SLFDn SLFDc1 SLFDc2
$ N $ $ L_1 $ error Order $ L_1 $ error Order $ L_1 $ error 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
Table 7.  Numerical $ L_\infty $ errors and orders for the nonlinear Burgers' equation (44). $ T = 0.5/\pi $ and $ \text{CFL} = 1.5 $ in (40). 4th order RKEI (3) is used
SLFDn SLFDc1 SLFDc2
$ N $ $ L_{\infty} $ error Order $ L_{\infty} $ error Order $ L_{\infty} $ error 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
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SLFDn SLFDc1 SLFDc2
$ N $ $ L_{\infty} $ error Order $ L_{\infty} $ error Order $ L_{\infty} $ error 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
Table 8.  Numerical errors and orders for the nonlinear Burgers' equation (44) from different CFL numbers, with SLFDn, 3rd order (2) and 4th order (3) in time, respectively. 4th order linear interpolation is used. $ N = 800 $ and $ T = 0.5/\pi $
CFL 3rd order SLFDn 4th order SLFDn
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
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CFL 3rd order SLFDn 4th order SLFDn
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
Table 9.  Numerical errors and orders for the nonlinear Burgers' equation (44) from different CFL numbers, with the two 3rd order RKEI conservative schemes. $ N = 800 $ and $ T = 0.5/\pi $
CFL 3rd order SLFDc2 3rd order SLFDc1
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
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CFL 3rd order SLFDc2 3rd order SLFDc1
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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
Table 10.  Numerical errors and orders for the nonlinear Burgers' equation (44) from different CFL numbers, with the two 4th order RKEI conservative schemes. $ N = 800 $ and $ T = 0.5/\pi $
CFL 4th order SLFDc2 4th order SLFDc1
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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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CFL 4th order SLFDc2 4th order SLFDc1
$ L_1 $ error Order $ L_{\infty} $ error Order $ L_1 $ error Order $ L_{\infty} $ error 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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