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December  2019, 14(4): 659-676. doi: 10.3934/nhm.2019026

Well-balanced scheme for gas-flow in pipeline networks

Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany

* Corresponding author: Yogiraj Mantri

Received  June 2018 Revised  May 2019 Published  October 2019

Gas flow through pipeline networks can be described using $ 2\times 2 $ hyperbolic balance laws along with coupling conditions at nodes. The numerical solution at steady state is highly sensitive to these coupling conditions and also to the balance between flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & Özcan[11] introduced a well-balanced method for general $ 2\times 2 $ systems of balance laws. In this paper, we simplify and extend this approach to a network of pipes. We prove well-balancing for different coupling conditions and for compressors stations, and demonstrate the advantage of the scheme by numerical experiments.

Citation: Yogiraj Mantri, Michael Herty, Sebastian Noelle. Well-balanced scheme for gas-flow in pipeline networks. Networks & Heterogeneous Media, 2019, 14 (4) : 659-676. doi: 10.3934/nhm.2019026
References:
[1]

E. AudusseF. BouchutM.-O. BristeauR. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), 2050-2065.  doi: 10.1137/S1064827503431090.  Google Scholar

[2]

M. K. BandaA.-S. Häck and M. Herty, Numerical discretization of coupling conditions by high-order schemes, J. Sci. Comput., 69 (2016), 122-145.  doi: 10.1007/s10915-016-0185-x.  Google Scholar

[3]

M. K. BandaM. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314.  doi: 10.3934/nhm.2006.1.295.  Google Scholar

[4]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[5]

A. BermúdezX. López and M. E. Vázquez-Cendón, Treating network junctions in finite volume solution of transient gas flow models, J. Comput. Phys., 344 (2017), 187-209.  doi: 10.1016/j.jcp.2017.04.066.  Google Scholar

[6]

A. BollermannG. X. ChenA. Kurganov and S. Noelle, A well-balanced reconstruction of wet/dry fronts for the shallow water equations, J. Sci. Comput., 56 (2013), 267-290.  doi: 10.1007/s10915-012-9677-5.  Google Scholar

[7]

R. Borsche and J. Kall, ADER schemes and high order coupling on networks of hyperbolic conservation laws, J. Comput. Phys., 273 (2014), 658-670.  doi: 10.1016/j.jcp.2014.05.042.  Google Scholar

[8]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[9]

J. BrouwerI. Gasser and M. Herty, Gas pipeline models revisited: Model hierarchies, nonisothermal models, and simulations of networks, Multiscale Model. Simul., 9 (2011), 601-623.  doi: 10.1137/100813580.  Google Scholar

[10]

G. X. Chen and S. Noelle, A new hydrostatic reconstruction scheme based on subcell reconstructions, SIAM J. Numer. Anal., 55 (2017), 758-784.  doi: 10.1137/15M1053074.  Google Scholar

[11]

A. ChertockM. Herty and Ş. N. Özcan, Well-balanced central-upwind schemes for $2\times 2$ system of balance laws, Theory, Numerics and Applications of Hyperbolic Problems. Ⅰ, Springer Proc. Math. Stat. Springer, Cham, 236 (2018), 345-361.   Google Scholar

[12]

R. M. ColomboG. GuerraM. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[13]

R. M. ColomboM. Herty and V. Sachers, On $2\times 2$ conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.  doi: 10.1137/070690298.  Google Scholar

[14]

R. M. Colombo and M. Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media, 1 (2006), 495-511.  doi: 10.3934/nhm.2006.1.495.  Google Scholar

[15]

R. M. Colombo and M. Garavello, On the Cauchy problem for the $p$-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.  doi: 10.1137/060665841.  Google Scholar

[16]

R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, J. Hyperbolic Differ. Equ., 5 (2008), 547-568.  doi: 10.1142/S0219891608001593.  Google Scholar

[17]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948.  Google Scholar

[18]

S. A. DyachenkoA. ZlotnikA. O. Korotkevich and M. Chertkov, Operator splitting method for simulation of dynamic flows in natural gas pipeline networks, Phys. D, 361 (2017), 1-11.  doi: 10.1016/j.physd.2017.09.002.  Google Scholar

[19]

H. Egger, A robust conservative mixed finite element method for isentropic compressible flow on pipe networks, SIAM J. Sci. Comput., 40 (2018), A108–A129. doi: 10.1137/16M1094373.  Google Scholar

[20]

E. GodlewskiK.-C. Le Thanh and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. Ⅱ. The case of systems, M2AN Math. Model. Numer. Anal., 39 (2005), 649-692.  doi: 10.1051/m2an:2005029.  Google Scholar

[21]

M. GugatM. Herty and S. Müller, Coupling conditions for the transition from supersonic to subsonic fluid states, Netw. Heterog. Media, 12 (2017), 371-380.  doi: 10.3934/nhm.2017016.  Google Scholar

[22]

M. Gugat and S. Ulbrich, The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, J. Math. Anal. Appl., 454 (2017), 439-452.  doi: 10.1016/j.jmaa.2017.04.064.  Google Scholar

[23]

M. HertyJ. Mohring and V. Sachers, A new model for gas flow in pipe networks, Math. Methods Appl. Sci., 33 (2010), 845-855.  doi: 10.1002/mma.1197.  Google Scholar

[24]

M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Internat. J. Numer. Methods Fluids, 56 (2008), 485-506.  doi: 10.1002/fld.1531.  Google Scholar

[25]

A. KurganovS. Noelle and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23 (2001), 707-740.  doi: 10.1137/S1064827500373413.  Google Scholar

[26]

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241-282.  doi: 10.1006/jcph.2000.6459.  Google Scholar

[27]

A. Morin and G. A. Reigstad, Pipe networks: Coupling constants in a junction for the isentropic euler equations, Energy Procedia, 64 (2015), 140-149.  doi: 10.1016/j.egypro.2015.01.017.  Google Scholar

[28]

A. NaumannO. Kolb and M. Semplice, On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws, Appl. Math. Comput., 325 (2018), 252-270.  doi: 10.1016/j.amc.2017.12.041.  Google Scholar

[29]

S. NoelleN. PankratzG. Puppo and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), 474-499.  doi: 10.1016/j.jcp.2005.08.019.  Google Scholar

[30]

S. NoelleY. L. Xing and C.-W. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys., 226 (2007), 29-58.  doi: 10.1016/j.jcp.2007.03.031.  Google Scholar

[31]

A. Osiadacz, Nonlinear programming applied to the optimum control of a gas compressor station, Internat. J. Numer. Methods Engrg., 15 (1980), 1287-1301.  doi: 10.1002/nme.1620150902.  Google Scholar

[32]

G. Puppo and G. Russo, Numerical Methods for Balance Laws, Quaderni di Matematica, 24. Department of Mathematics, Seconda Università di Napoli, Caserta, 2009.  Google Scholar

[33]

G. A. Reigstad, Numerical network models and entropy principles for isothermal junction flow, Netw. Heterog. Media, 9 (2014), 65-95.  doi: 10.3934/nhm.2014.9.65.  Google Scholar

[34]

G. A. Reigstad, Existence and uniqueness of solutions to the generalized {R}iemann problem for isentropic flow, SIAM J. Appl. Math., 75 (2015), 679-702.  doi: 10.1137/140962759.  Google Scholar

[35]

G. A. ReigstadT. FlåttenN. Erland Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbolic Differ. Equ., 12 (2015), 37-59.  doi: 10.1142/S0219891615500022.  Google Scholar

show all references

References:
[1]

E. AudusseF. BouchutM.-O. BristeauR. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), 2050-2065.  doi: 10.1137/S1064827503431090.  Google Scholar

[2]

M. K. BandaA.-S. Häck and M. Herty, Numerical discretization of coupling conditions by high-order schemes, J. Sci. Comput., 69 (2016), 122-145.  doi: 10.1007/s10915-016-0185-x.  Google Scholar

[3]

M. K. BandaM. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314.  doi: 10.3934/nhm.2006.1.295.  Google Scholar

[4]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[5]

A. BermúdezX. López and M. E. Vázquez-Cendón, Treating network junctions in finite volume solution of transient gas flow models, J. Comput. Phys., 344 (2017), 187-209.  doi: 10.1016/j.jcp.2017.04.066.  Google Scholar

[6]

A. BollermannG. X. ChenA. Kurganov and S. Noelle, A well-balanced reconstruction of wet/dry fronts for the shallow water equations, J. Sci. Comput., 56 (2013), 267-290.  doi: 10.1007/s10915-012-9677-5.  Google Scholar

[7]

R. Borsche and J. Kall, ADER schemes and high order coupling on networks of hyperbolic conservation laws, J. Comput. Phys., 273 (2014), 658-670.  doi: 10.1016/j.jcp.2014.05.042.  Google Scholar

[8]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[9]

J. BrouwerI. Gasser and M. Herty, Gas pipeline models revisited: Model hierarchies, nonisothermal models, and simulations of networks, Multiscale Model. Simul., 9 (2011), 601-623.  doi: 10.1137/100813580.  Google Scholar

[10]

G. X. Chen and S. Noelle, A new hydrostatic reconstruction scheme based on subcell reconstructions, SIAM J. Numer. Anal., 55 (2017), 758-784.  doi: 10.1137/15M1053074.  Google Scholar

[11]

A. ChertockM. Herty and Ş. N. Özcan, Well-balanced central-upwind schemes for $2\times 2$ system of balance laws, Theory, Numerics and Applications of Hyperbolic Problems. Ⅰ, Springer Proc. Math. Stat. Springer, Cham, 236 (2018), 345-361.   Google Scholar

[12]

R. M. ColomboG. GuerraM. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[13]

R. M. ColomboM. Herty and V. Sachers, On $2\times 2$ conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622.  doi: 10.1137/070690298.  Google Scholar

[14]

R. M. Colombo and M. Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media, 1 (2006), 495-511.  doi: 10.3934/nhm.2006.1.495.  Google Scholar

[15]

R. M. Colombo and M. Garavello, On the Cauchy problem for the $p$-system at a junction, SIAM J. Math. Anal., 39 (2008), 1456-1471.  doi: 10.1137/060665841.  Google Scholar

[16]

R. M. Colombo and C. Mauri, Euler system for compressible fluids at a junction, J. Hyperbolic Differ. Equ., 5 (2008), 547-568.  doi: 10.1142/S0219891608001593.  Google Scholar

[17]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948.  Google Scholar

[18]

S. A. DyachenkoA. ZlotnikA. O. Korotkevich and M. Chertkov, Operator splitting method for simulation of dynamic flows in natural gas pipeline networks, Phys. D, 361 (2017), 1-11.  doi: 10.1016/j.physd.2017.09.002.  Google Scholar

[19]

H. Egger, A robust conservative mixed finite element method for isentropic compressible flow on pipe networks, SIAM J. Sci. Comput., 40 (2018), A108–A129. doi: 10.1137/16M1094373.  Google Scholar

[20]

E. GodlewskiK.-C. Le Thanh and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. Ⅱ. The case of systems, M2AN Math. Model. Numer. Anal., 39 (2005), 649-692.  doi: 10.1051/m2an:2005029.  Google Scholar

[21]

M. GugatM. Herty and S. Müller, Coupling conditions for the transition from supersonic to subsonic fluid states, Netw. Heterog. Media, 12 (2017), 371-380.  doi: 10.3934/nhm.2017016.  Google Scholar

[22]

M. Gugat and S. Ulbrich, The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, J. Math. Anal. Appl., 454 (2017), 439-452.  doi: 10.1016/j.jmaa.2017.04.064.  Google Scholar

[23]

M. HertyJ. Mohring and V. Sachers, A new model for gas flow in pipe networks, Math. Methods Appl. Sci., 33 (2010), 845-855.  doi: 10.1002/mma.1197.  Google Scholar

[24]

M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Internat. J. Numer. Methods Fluids, 56 (2008), 485-506.  doi: 10.1002/fld.1531.  Google Scholar

[25]

A. KurganovS. Noelle and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23 (2001), 707-740.  doi: 10.1137/S1064827500373413.  Google Scholar

[26]

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), 241-282.  doi: 10.1006/jcph.2000.6459.  Google Scholar

[27]

A. Morin and G. A. Reigstad, Pipe networks: Coupling constants in a junction for the isentropic euler equations, Energy Procedia, 64 (2015), 140-149.  doi: 10.1016/j.egypro.2015.01.017.  Google Scholar

[28]

A. NaumannO. Kolb and M. Semplice, On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws, Appl. Math. Comput., 325 (2018), 252-270.  doi: 10.1016/j.amc.2017.12.041.  Google Scholar

[29]

S. NoelleN. PankratzG. Puppo and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), 474-499.  doi: 10.1016/j.jcp.2005.08.019.  Google Scholar

[30]

S. NoelleY. L. Xing and C.-W. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys., 226 (2007), 29-58.  doi: 10.1016/j.jcp.2007.03.031.  Google Scholar

[31]

A. Osiadacz, Nonlinear programming applied to the optimum control of a gas compressor station, Internat. J. Numer. Methods Engrg., 15 (1980), 1287-1301.  doi: 10.1002/nme.1620150902.  Google Scholar

[32]

G. Puppo and G. Russo, Numerical Methods for Balance Laws, Quaderni di Matematica, 24. Department of Mathematics, Seconda Università di Napoli, Caserta, 2009.  Google Scholar

[33]

G. A. Reigstad, Numerical network models and entropy principles for isothermal junction flow, Netw. Heterog. Media, 9 (2014), 65-95.  doi: 10.3934/nhm.2014.9.65.  Google Scholar

[34]

G. A. Reigstad, Existence and uniqueness of solutions to the generalized {R}iemann problem for isentropic flow, SIAM J. Appl. Math., 75 (2015), 679-702.  doi: 10.1137/140962759.  Google Scholar

[35]

G. A. ReigstadT. FlåttenN. Erland Haugen and T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow, J. Hyperbolic Differ. Equ., 12 (2015), 37-59.  doi: 10.1142/S0219891615500022.  Google Scholar

Figure 1.  Intersection of three pipes at junction O. Right-Zoomed view of the junction with old traces $ U_{{i}}^{{o}} $ and new traces $ U_{{i}}^{{*}} $ given in Section 2
Figure 2.  Phase plot in terms of equilibrium variables with initial state $ V_i^o = (0.1, 0.4)^T $
Figure 3.  Momentum for perturbation of order $ 10^{-3} $ for a node connected to two pipes
Figure 4.  Momentum for perturbation of order $ 10^{-6} $ for a node connected to two pipes
Figure 5.  Momentum for perturbation of order $ 10^{-3} $ for a node connected to one incoming and two outgoing pipes
Figure 6.  Momentum for perturbation of order $ 10^{-6} $ for a node connected to one incoming and two outgoing pipes
Figure 7.  Conservative variables, $ \rho, q $ at T = 0.1 in pipes 1, 2, 3 with WB and NWB scheme
Figure 8.  Conservative variables, $ \rho, q $ at T = 0.25 in pipes 1, 2, 3 with WB and NWB scheme
Table 1.  Comparison of L-1 errors between well-balanced(WB) and non well-balanced(NWB) scheme at steady state for a junction at time T = 1
No. of cells in each pipe L1-error for variable 1 Incoming, 1 Outgoing 1 Incoming, 2 Outgoing 2 Incoming, 1 Outgoing
WBNWBWBNWBWBNWB
50K$2.83\text{x}10^{-17}$$6.19\text{x}10^{-7}$ $6.91\text{x}10^{-17}$$3.78\text{x}10^{-7}$ $9.02\text{x}10^{-17}$$3.45\text{x}10^{-7}$
L$3.44\text{x}10^{-17}$$9.48\text{x}10^{-7}$$5.16\text{x}10^{-17}$$3.57\text{x}10^{-7}$$9.21\text{x}10^{-17}$$7.38\text{x}10^{-7}$
100K$3.95\text{x}10^{-17}$$1.56\text{x}10^{-7}$$8.12\text{x}10^{-17}$$9.63\text{x}10^{-8}$$8.60\text{x}10^{-17}$$8.67\text{x}10^{-8}$
L$4.86\text{x}10^{-17}$$2.43\text{x}10^{-7}$$7.38\text{x}10^{-17}$$8.94\text{x}10^{-8}$$8.24\text{x}10^{-17}$$1.87\text{x}10^{-7}$
200K$5.11\text{x}10^{-17}$$3.88\text{x}10^{-8}$ $8.69\text{x}10^{-17}$$2.62\text{x}10^{-8}$ $1.04\text{x}10^{-16}$$2.69\text{x}10^{-8}$
L$5.85\text{x}10^{-17}$$6.13\text{x}10^{-8}$$7.06\text{x}10^{-17}$$2.32\text{x}10^{-8}$$9.49\text{x}10^{-17}$$5.03\text{x}10^{-8}$
No. of cells in each pipe L1-error for variable 1 Incoming, 1 Outgoing 1 Incoming, 2 Outgoing 2 Incoming, 1 Outgoing
WBNWBWBNWBWBNWB
50K$2.83\text{x}10^{-17}$$6.19\text{x}10^{-7}$ $6.91\text{x}10^{-17}$$3.78\text{x}10^{-7}$ $9.02\text{x}10^{-17}$$3.45\text{x}10^{-7}$
L$3.44\text{x}10^{-17}$$9.48\text{x}10^{-7}$$5.16\text{x}10^{-17}$$3.57\text{x}10^{-7}$$9.21\text{x}10^{-17}$$7.38\text{x}10^{-7}$
100K$3.95\text{x}10^{-17}$$1.56\text{x}10^{-7}$$8.12\text{x}10^{-17}$$9.63\text{x}10^{-8}$$8.60\text{x}10^{-17}$$8.67\text{x}10^{-8}$
L$4.86\text{x}10^{-17}$$2.43\text{x}10^{-7}$$7.38\text{x}10^{-17}$$8.94\text{x}10^{-8}$$8.24\text{x}10^{-17}$$1.87\text{x}10^{-7}$
200K$5.11\text{x}10^{-17}$$3.88\text{x}10^{-8}$ $8.69\text{x}10^{-17}$$2.62\text{x}10^{-8}$ $1.04\text{x}10^{-16}$$2.69\text{x}10^{-8}$
L$5.85\text{x}10^{-17}$$6.13\text{x}10^{-8}$$7.06\text{x}10^{-17}$$2.32\text{x}10^{-8}$$9.49\text{x}10^{-17}$$5.03\text{x}10^{-8}$
Table 2.  Comparison of L-1 errors between well-balanced(WB) and non well-balanced(NWB) scheme at steady state with a compressor at different compression ratios at time T = 1
No. of cells in each pipe L1-error for variable CR=1.5 CR=2.0 CR=2.5
WB NWB WB NWB WB NWB
50 K $1.11\text{x}10^{-17}$ $4.16\text{x}10^{-7}$ $5.30\text{x}10^{-17}$ $3.78\text{x}10^{-7}$ $1.97\text{x}10^{-17}$ $3.77\text{x}10^{-7}$
L $2.66\text{x}10^{-17}$ $4.00\text{x}10^{-7}$ $5.38\text{x}10^{-17}$ $3.57\text{x}10^{-7}$ $1.39\text{x}10^{-17}$ $3.54\text{x}10^{-7}$
100 K $2.90\text{x}10^{-17}$ $1.05\text{x}10^{-7}$ $7.28\text{x}10^{-17}$ $9.63\text{x}10^{-8}$ $4.22\text{x}10^{-17}$ $9.68\text{x}10^{-8}$
L $4.08\text{x}10^{-17}$ $1.01\text{x}10^{-7}$ $7.24\text{x}10^{-17}$ $8.94\text{x}10^{-8}$ $4.66\text{x}10^{-17}$ $8.89\text{x}10^{-7}$
200 K $4.26\text{x}10^{-17}$ $2.64\text{x}10^{-8}$ $8.15\text{x}10^{-17}$ $2.62\text{x}10^{-8}$ $5.02\text{x}10^{-17}$ $2.84\text{x}10^{-8}$
L $4.69\text{x}10^{-17}$ $2.53\text{x}10^{-8}$ $7.45\text{x}10^{-17}$ $2.32\text{x}10^{-8}$ $5.76\text{x}10^{-17}$ $2.59\text{x}10^{-8}$
No. of cells in each pipe L1-error for variable CR=1.5 CR=2.0 CR=2.5
WB NWB WB NWB WB NWB
50 K $1.11\text{x}10^{-17}$ $4.16\text{x}10^{-7}$ $5.30\text{x}10^{-17}$ $3.78\text{x}10^{-7}$ $1.97\text{x}10^{-17}$ $3.77\text{x}10^{-7}$
L $2.66\text{x}10^{-17}$ $4.00\text{x}10^{-7}$ $5.38\text{x}10^{-17}$ $3.57\text{x}10^{-7}$ $1.39\text{x}10^{-17}$ $3.54\text{x}10^{-7}$
100 K $2.90\text{x}10^{-17}$ $1.05\text{x}10^{-7}$ $7.28\text{x}10^{-17}$ $9.63\text{x}10^{-8}$ $4.22\text{x}10^{-17}$ $9.68\text{x}10^{-8}$
L $4.08\text{x}10^{-17}$ $1.01\text{x}10^{-7}$ $7.24\text{x}10^{-17}$ $8.94\text{x}10^{-8}$ $4.66\text{x}10^{-17}$ $8.89\text{x}10^{-7}$
200 K $4.26\text{x}10^{-17}$ $2.64\text{x}10^{-8}$ $8.15\text{x}10^{-17}$ $2.62\text{x}10^{-8}$ $5.02\text{x}10^{-17}$ $2.84\text{x}10^{-8}$
L $4.69\text{x}10^{-17}$ $2.53\text{x}10^{-8}$ $7.45\text{x}10^{-17}$ $2.32\text{x}10^{-8}$ $5.76\text{x}10^{-17}$ $2.59\text{x}10^{-8}$
[1]

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