December  2020, 15(4): 653-679. doi: 10.3934/nhm.2020018

Efficient numerical methods for gas network modeling and simulation

1. 

School of Information Science and Technology, ShanghaiTech University, 393 Middle Huaxia Road, 201210, Shanghai, China

2. 

Computational Methods for Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106, Magdeburg, Germany

3. 

Faculty of Mathematics, Technische Universität Chemnitz, Reichenhainer Straße 41, 09126, Chemnitz, Germany

* Corresponding author: Yue Qiu

Received  August 2019 Revised  August 2020 Published  August 2020

Fund Project: This work is partially funded by the European Regional Development Fund (ERDF/EFRE: ZS/2016/04/78156) within the Center Dynamic Systems (CDS)

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a system of nonlinear differential algebraic equations (DAEs). With our modeling approach, we reduce the number of algebraic constraints, which correspond to the $ (2,2) $ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $ (1, 1) $ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

Citation: Yue Qiu, Sara Grundel, Martin Stoll, Peter Benner. Efficient numerical methods for gas network modeling and simulation. Networks & Heterogeneous Media, 2020, 15 (4) : 653-679. doi: 10.3934/nhm.2020018
References:
[1]

N. Banagaaya, S. Grundel and P. Benner, Index-aware MOR for gas transport networks with many supply inputs, in IUTAM Symposium on Model Order Reduction of Coupled Systems (eds. J. Fehr and B. Haasdonk), Springer International Publishing, Cham, 2020,191–207. Google Scholar

[2]

J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-998-1.  Google Scholar

[3]

P. Benner, S. Grundel, C. Himpe, C. Huck, T. Streubel and C. Tischendorf, Gas network benchmark models, in Differential-Algebraic Equations Forum, Springer, Berlin, Heidelberg, 2018.  Google Scholar

[4]

M. BenziG. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137.  doi: 10.1017/S0962492904000212.  Google Scholar

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A. BermúdezX. López and M. E. Vázquez-Cendón, Finite volume methods for multi-component Euler equations with source terms, Comput. Fluids, 156 (2017), 113-134.  doi: 10.1016/j.compfluid.2017.07.004.  Google Scholar

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M. Chaczykowski, Sensitivity of pipeline gas flow model to the selection of the equation of state, Chem. Eng. Res. Des., 87 (2009), 1596-1603.  doi: 10.1016/j.cherd.2009.06.008.  Google Scholar

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R. DemboS. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408.  doi: 10.1137/0719025.  Google Scholar

[8]

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

[9]

H. Egger, T. Kugler and N. Strogies, Parameter identification in a semilinear hyperbolic system, Inverse Probl., 33 (2017), 055022. doi: 10.1088/1361-6420/aa648c.  Google Scholar

[10] H. ElmanD. Silvester and A. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, New York, 2014.  doi: 10.1093/acprof:oso/9780199678792.001.0001.  Google Scholar
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A. Fügenschuh, et al., Physical and technical fundamentals of gas networks, in Evaluating Gas Network Capacities Google Scholar

[12]

T. G. GrandónH. Heitsch and R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14 (2017), 443-460.  doi: 10.1007/s10287-017-0284-7.  Google Scholar

[13]

S. Grundel, N. Hornung, B. Klaassen, P. Benner and T. Clees, Computing surrogates for gas network simulation using model order reduction, in Surrogate-Based Modeling and Optimization doi: 10.1007/978-1-4614-7551-4_9.  Google Scholar

[14]

S. Grundel, N. Hornung and S. Roggendorf, Numerical aspects of model order reduction for gas transportation networks, in Simulation-Driven Modeling and Optimization, Springer Proceedings in Mathematics & Statistics, 153, 2016, 1–28.  Google Scholar

[15]

S. Grundel and L. Jansen, Efficient simulation of transient gas networks using IMEX integration schemes and MOR methods, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4579–4584. doi: 10.1109/CDC.2015.7402934.  Google Scholar

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S. Grundel, L. Jansen, N. Hornung, T. Clees, C. Tischendorf and P. Benner, Model order reduction of differential algebraic equations arising from the simulation of gas transport networks, in Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, Springer Berlin Heidelberg, 2014,183–205. doi: 10.1007/978-3-642-34928-7_2.  Google Scholar

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F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications, Springer Verlag, Singapore, 2017, 77–122.  Google Scholar

[19]

A. Herrán-GonzálezJ. M. D. L. CruzB. D. Andrés-Toro and J. L. Risco-Martín, Modeling and simulation of a gas distribution pipeline network, Appl. Math. Model., 33 (2009), 1584-1600.   Google Scholar

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M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), 81-97.  doi: 10.3934/nhm.2007.2.81.  Google Scholar

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M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612.  doi: 10.1137/070688535.  Google Scholar

[22]

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

[23]

C. HuckL. Jansen and C. Tischendorf, A topology based discretization of PDAEs describing water transportation networks, Proc. Appl. Math. Mech., 14 (2014), 923-924.  doi: 10.1002/pamm.201410442.  Google Scholar

[24]

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26.  doi: 10.1090/S0025-5718-1986-0815828-4.  Google Scholar

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C. Kelley, Solving Nonlinear Equations with Newton's Method, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718898.  Google Scholar

[26]

A. Osiadacz, Simulation of transient gas flows in networks, Internat. J. Numer. Methods Fluids, 4 (1984), 13-24.  doi: 10.1002/fld.1650040103.  Google Scholar

[27]

A. Osiadacz, Simulation and Analysis of Gas Networks, Gulf Publishing, Houston, TX, 1987. Google Scholar

[28]

A. J. Osiadacz and M. Yedroudj, A comparison of a finite element method and a finite difference method for transient simulation of a gas pipeline, Appl. Math. Model., 13 (1989), 79-85.  doi: 10.1016/0307-904X(89)90018-8.  Google Scholar

[29]

J. W. Pearson, On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems, Electron. Trans. Numer. Anal., 44 (2015), 53-72.   Google Scholar

[30]

J. Pestana and A. J. Wathen, Natural preconditioning and iterative methods for saddle point systems, SIAM Rev., 57 (2015), 71-91.  doi: 10.1137/130934921.  Google Scholar

[31]

M. Porcelli, V. Simoncini and M. Tani, Preconditioning of active-set Newton methods for PDE-constrained optimal control problems, SIAM J. Sci. Comput., 37 (2015), S472–S502. doi: 10.1137/140975711.  Google Scholar

[32]

Y. Qiu, Preconditioning Optimal Flow Control Problems Using Multilevel Sequentially Semiseparable Matrix Computations, Ph.D thesis, Delft Institute of Applied Mathematics, Delft University of Technology, 2015. Google Scholar

[33]

T. Rees, Preconditioning Iterative Methods for PDE-Constrained Optimization, Ph.D thesis, University of Oxford, 2010. Google Scholar

[34]

S. Roggendorf, Model Order Reduction for Linearized Systems Arising from the Simulation of Gas Transportation Networks, Master's thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn, Germany, 2015. Google Scholar

[35]

Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[36]

P. Sonneveld and M. B. van Gijzen, IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31 (2008), 1035-1062.  doi: 10.1137/070685804.  Google Scholar

[37]

M. C. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361.  doi: 10.1016/j.cam.2006.04.018.  Google Scholar

[38]

M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM J. Sci. Comput., 37 (2015), B1–B29. doi: 10.1137/130926365.  Google Scholar

[39]

W. Q. Tao and H. C. Ti, Transient analysis of gas pipeline network, Chem. Eng. J., 69 (1998), 47-52.  doi: 10.1016/S1385-8947(97)00109-5.  Google Scholar

[40]

E. F. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20 (2000), 47-79.  doi: 10.1093/imanum/20.1.47.  Google Scholar

[41]

M. B. van Gijzen and P. Sonneveld, Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties, ACM Trans. Math. Software, 38 (2011), 5: 1–5: 19. doi: 10.1145/2049662.2049667.  Google Scholar

[42]

A. J. Wathen, Preconditioning, Acta Numer., 24 (2015), 329-376.  doi: 10.1017/S0962492915000021.  Google Scholar

[43]

M. Wathen, C. Greif and D. Schötzau, Preconditioners for mixed finite element discretizations of incompressible MHD equations, SIAM J. Sci. Comput., 39 (2017), A2993–A3013. doi: 10.1137/16M1098991.  Google Scholar

[44]

J. Zhou and M. A. Adewumi, Simulation of transients in natural gas pipelines using hybrid TVD schemes, Internat. J. Numer. Methods Fluids, 32 (2000), 407-437.  doi: 10.1002/(SICI)1097-0363(20000229)32:4<407::AID-FLD945>3.0.CO;2-9.  Google Scholar

[45]

A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4563–4570. doi: 10.1109/TCNS.2014.2367360.  Google Scholar

show all references

References:
[1]

N. Banagaaya, S. Grundel and P. Benner, Index-aware MOR for gas transport networks with many supply inputs, in IUTAM Symposium on Model Order Reduction of Coupled Systems (eds. J. Fehr and B. Haasdonk), Springer International Publishing, Cham, 2020,191–207. Google Scholar

[2]

J. Bang-Jensen and G. Z. Gutin, Digraphs: Theory, Algorithms and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84800-998-1.  Google Scholar

[3]

P. Benner, S. Grundel, C. Himpe, C. Huck, T. Streubel and C. Tischendorf, Gas network benchmark models, in Differential-Algebraic Equations Forum, Springer, Berlin, Heidelberg, 2018.  Google Scholar

[4]

M. BenziG. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137.  doi: 10.1017/S0962492904000212.  Google Scholar

[5]

A. BermúdezX. López and M. E. Vázquez-Cendón, Finite volume methods for multi-component Euler equations with source terms, Comput. Fluids, 156 (2017), 113-134.  doi: 10.1016/j.compfluid.2017.07.004.  Google Scholar

[6]

M. Chaczykowski, Sensitivity of pipeline gas flow model to the selection of the equation of state, Chem. Eng. Res. Des., 87 (2009), 1596-1603.  doi: 10.1016/j.cherd.2009.06.008.  Google Scholar

[7]

R. DemboS. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408.  doi: 10.1137/0719025.  Google Scholar

[8]

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

[9]

H. Egger, T. Kugler and N. Strogies, Parameter identification in a semilinear hyperbolic system, Inverse Probl., 33 (2017), 055022. doi: 10.1088/1361-6420/aa648c.  Google Scholar

[10] H. ElmanD. Silvester and A. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, New York, 2014.  doi: 10.1093/acprof:oso/9780199678792.001.0001.  Google Scholar
[11]

A. Fügenschuh, et al., Physical and technical fundamentals of gas networks, in Evaluating Gas Network Capacities Google Scholar

[12]

T. G. GrandónH. Heitsch and R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Comput. Manag. Sci., 14 (2017), 443-460.  doi: 10.1007/s10287-017-0284-7.  Google Scholar

[13]

S. Grundel, N. Hornung, B. Klaassen, P. Benner and T. Clees, Computing surrogates for gas network simulation using model order reduction, in Surrogate-Based Modeling and Optimization doi: 10.1007/978-1-4614-7551-4_9.  Google Scholar

[14]

S. Grundel, N. Hornung and S. Roggendorf, Numerical aspects of model order reduction for gas transportation networks, in Simulation-Driven Modeling and Optimization, Springer Proceedings in Mathematics & Statistics, 153, 2016, 1–28.  Google Scholar

[15]

S. Grundel and L. Jansen, Efficient simulation of transient gas networks using IMEX integration schemes and MOR methods, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4579–4584. doi: 10.1109/CDC.2015.7402934.  Google Scholar

[16]

S. Grundel, L. Jansen, N. Hornung, T. Clees, C. Tischendorf and P. Benner, Model order reduction of differential algebraic equations arising from the simulation of gas transport networks, in Progress in Differential-Algebraic Equations, Differential-Algebraic Equations Forum, Springer Berlin Heidelberg, 2014,183–205. doi: 10.1007/978-3-642-34928-7_2.  Google Scholar

[17]

M. GugatF. M. HanteM. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320.  doi: 10.3934/nhm.2015.10.295.  Google Scholar

[18]

F. M. Hante, G. Leugering, A. Martin, L. Schewe and M. Schmidt, Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications, Springer Verlag, Singapore, 2017, 77–122.  Google Scholar

[19]

A. Herrán-GonzálezJ. M. D. L. CruzB. D. Andrés-Toro and J. L. Risco-Martín, Modeling and simulation of a gas distribution pipeline network, Appl. Math. Model., 33 (2009), 1584-1600.   Google Scholar

[20]

M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), 81-97.  doi: 10.3934/nhm.2007.2.81.  Google Scholar

[21]

M. Herty, Coupling conditions for networked systems of Euler equations, SIAM J. Sci. Comput., 30 (2008), 1596-1612.  doi: 10.1137/070688535.  Google Scholar

[22]

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

[23]

C. HuckL. Jansen and C. Tischendorf, A topology based discretization of PDAEs describing water transportation networks, Proc. Appl. Math. Mech., 14 (2014), 923-924.  doi: 10.1002/pamm.201410442.  Google Scholar

[24]

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26.  doi: 10.1090/S0025-5718-1986-0815828-4.  Google Scholar

[25]

C. Kelley, Solving Nonlinear Equations with Newton's Method, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718898.  Google Scholar

[26]

A. Osiadacz, Simulation of transient gas flows in networks, Internat. J. Numer. Methods Fluids, 4 (1984), 13-24.  doi: 10.1002/fld.1650040103.  Google Scholar

[27]

A. Osiadacz, Simulation and Analysis of Gas Networks, Gulf Publishing, Houston, TX, 1987. Google Scholar

[28]

A. J. Osiadacz and M. Yedroudj, A comparison of a finite element method and a finite difference method for transient simulation of a gas pipeline, Appl. Math. Model., 13 (1989), 79-85.  doi: 10.1016/0307-904X(89)90018-8.  Google Scholar

[29]

J. W. Pearson, On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems, Electron. Trans. Numer. Anal., 44 (2015), 53-72.   Google Scholar

[30]

J. Pestana and A. J. Wathen, Natural preconditioning and iterative methods for saddle point systems, SIAM Rev., 57 (2015), 71-91.  doi: 10.1137/130934921.  Google Scholar

[31]

M. Porcelli, V. Simoncini and M. Tani, Preconditioning of active-set Newton methods for PDE-constrained optimal control problems, SIAM J. Sci. Comput., 37 (2015), S472–S502. doi: 10.1137/140975711.  Google Scholar

[32]

Y. Qiu, Preconditioning Optimal Flow Control Problems Using Multilevel Sequentially Semiseparable Matrix Computations, Ph.D thesis, Delft Institute of Applied Mathematics, Delft University of Technology, 2015. Google Scholar

[33]

T. Rees, Preconditioning Iterative Methods for PDE-Constrained Optimization, Ph.D thesis, University of Oxford, 2010. Google Scholar

[34]

S. Roggendorf, Model Order Reduction for Linearized Systems Arising from the Simulation of Gas Transportation Networks, Master's thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn, Germany, 2015. Google Scholar

[35]

Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[36]

P. Sonneveld and M. B. van Gijzen, IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput., 31 (2008), 1035-1062.  doi: 10.1137/070685804.  Google Scholar

[37]

M. C. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361.  doi: 10.1016/j.cam.2006.04.018.  Google Scholar

[38]

M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM J. Sci. Comput., 37 (2015), B1–B29. doi: 10.1137/130926365.  Google Scholar

[39]

W. Q. Tao and H. C. Ti, Transient analysis of gas pipeline network, Chem. Eng. J., 69 (1998), 47-52.  doi: 10.1016/S1385-8947(97)00109-5.  Google Scholar

[40]

E. F. Toro and S. J. Billett, Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20 (2000), 47-79.  doi: 10.1093/imanum/20.1.47.  Google Scholar

[41]

M. B. van Gijzen and P. Sonneveld, Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties, ACM Trans. Math. Software, 38 (2011), 5: 1–5: 19. doi: 10.1145/2049662.2049667.  Google Scholar

[42]

A. J. Wathen, Preconditioning, Acta Numer., 24 (2015), 329-376.  doi: 10.1017/S0962492915000021.  Google Scholar

[43]

M. Wathen, C. Greif and D. Schötzau, Preconditioners for mixed finite element discretizations of incompressible MHD equations, SIAM J. Sci. Comput., 39 (2017), A2993–A3013. doi: 10.1137/16M1098991.  Google Scholar

[44]

J. Zhou and M. A. Adewumi, Simulation of transients in natural gas pipelines using hybrid TVD schemes, Internat. J. Numer. Methods Fluids, 32 (2000), 407-437.  doi: 10.1002/(SICI)1097-0363(20000229)32:4<407::AID-FLD945>3.0.CO;2-9.  Google Scholar

[45]

A. Zlotnik, M. Chertkov and S. Backhaus, Optimal control of transient flow in natural gas networks, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, 4563–4570. doi: 10.1109/TCNS.2014.2367360.  Google Scholar

Figure 1.  Separation of control volume $ C_i $
Figure 2.  A typical gas network
Figure 3.  Smoothed network of Figure 2 with an ordering of the pipes
Figure 7.  Computational diagram for gas network simulation
Figure 4.  An illustrative network example of a DAG
Figure 5.  Big benchmark network in [34]
Figure 6.  Sparsity pattern of $ J $ without and with DF ordering
Figure 8.  Pipeline network in [16]
Figure 9.  Comparison of FVM and FDM for a single pipe network
Figure 10.  Medium size network
Figure 11.  Comparison of FVM and FDM for a medium network
Figure 12.  Mass flow at supply nodes for case 1
Figure 13.  Mass flow at supply nodes for case 2
Figure 14.  Mass flow for the pipe $ 31\rightarrow 37 $
Figure 15.  Nonlinear residual at the first and tenth time step
Figure 16.  Number of IDR($ 4 $) iterations at the first time step
Figure 17.  Number of IDR($ 4 $) iterations at the $ 10 $-th time step
Algorithm 1: Newton's method to solve (16)
1: Input: maximal number of Newton steps $ n_{\max} $, stop tolerance $ \varepsilon_0 $, initial guess $ x_0 $
2: $ m=0 $
3: while $ m\leq n_{\max}\& \ \|F(x)\|\geq \varepsilon_0 $ do
4:     Compute the Jacobian matrix $ D_F(x_m)=\frac{\partial}{\partial x}F|_{x=x_m} $
5:     Solve $ F(x_m) + D_F(x_m)(x-x_m)=0 $
6:     $ m\gets m+1 $, $ x_m\gets x $
7: end while
8: Output: solution $ x\approx x_m $
Algorithm 1: Newton's method to solve (16)
1: Input: maximal number of Newton steps $ n_{\max} $, stop tolerance $ \varepsilon_0 $, initial guess $ x_0 $
2: $ m=0 $
3: while $ m\leq n_{\max}\& \ \|F(x)\|\geq \varepsilon_0 $ do
4:     Compute the Jacobian matrix $ D_F(x_m)=\frac{\partial}{\partial x}F|_{x=x_m} $
5:     Solve $ F(x_m) + D_F(x_m)(x-x_m)=0 $
6:     $ m\gets m+1 $, $ x_m\gets x $
7: end while
8: Output: solution $ x\approx x_m $
Table 1.  Computational time (seconds) for Schur complement $ S^1 $
$ h $ $ \# D_F $ with DF without DF
20 2.01e+05 $ 8.12 $ 8.75
10 3.97e+05 $ 17.84 $ 19.14
5 7.91e+05 $ 38.44 $ 41.75
2.5 1.58e+06 $ 81.42 $ 87.77
$ h $ $ \# D_F $ with DF without DF
20 2.01e+05 $ 8.12 $ 8.75
10 3.97e+05 $ 17.84 $ 19.14
5 7.91e+05 $ 38.44 $ 41.75
2.5 1.58e+06 $ 81.42 $ 87.77
Table 2.  Condition number of the Jacobian matrix $ D_{F} $ from FVM and FDM, 1st time step, $ h = 60 $
Newton iter. 1 2 3 4
FVM 1.56e+07 1.57e+07 1.57e+07 1.57e+07
FDM 1.24e+08 1.25e+08 1.25e+08 1.25e+08
Newton iter. 1 2 3 4
FVM 1.56e+07 1.57e+07 1.57e+07 1.57e+07
FDM 1.24e+08 1.25e+08 1.25e+08 1.25e+08
Table 3.  Computational time for the 1st Newton iteration
$ h $ $ \# D_F $ $ t_{S^1} $ IDR($ 4 $) backslash
40 1.03e+05 $ 3.85 $ 0.25 0.13
20 2.01e+05 $ 8.12 $ 0.52 0.36
10 3.97e+05 $ 17.84 $ 1.06 1.18
5 7.91e+05 $ 38.44 $ 2.13 1054.62
2.5 1.58e+06 $ 81.42 $ 4.34 -
$ h $ $ \# D_F $ $ t_{S^1} $ IDR($ 4 $) backslash
40 1.03e+05 $ 3.85 $ 0.25 0.13
20 2.01e+05 $ 8.12 $ 0.52 0.36
10 3.97e+05 $ 17.84 $ 1.06 1.18
5 7.91e+05 $ 38.44 $ 2.13 1054.62
2.5 1.58e+06 $ 81.42 $ 4.34 -
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