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2019, Volume 14, Issue 4: 659-676. Doi: 10.3934/nhm.2019026

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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
^* Corresponding author: Yogiraj Mantri

Received: June 2018

Revised: May 2019

Published: December 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L60, 35L65, 76M12.

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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

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Figure 2. Phase plot in terms of equilibrium variables with initial state $ V_i^o = (0.1, 0.4)^T $

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Figure 3. Momentum for perturbation of order $ 10^{-3} $ for a node connected to two pipes

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Figure 4. Momentum for perturbation of order $ 10^{-6} $ for a node connected to two pipes

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Figure 5. Momentum for perturbation of order $ 10^{-3} $ for a node connected to one incoming and two outgoing pipes

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Figure 6. Momentum for perturbation of order $ 10^{-6} $ for a node connected to one incoming and two outgoing pipes

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Figure 7. Conservative variables, $ \rho, q $ at T = 0.1 in pipes 1, 2, 3 with WB and NWB scheme

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Figure 8. Conservative variables, $ \rho, q $ at T = 0.25 in pipes 1, 2, 3 with WB and NWB scheme

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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
No. of cells in each pipe	L1-error for variable	WB	NWB	WB	NWB	WB	NWB
50	K	$2.83\text{x}10^{-17}$	$6.19\text{x}10^{-7}$	$6.91\text{x}10^{-17}$	$3.78\text{x}10^{-7}$	$9.02\text{x}10^{-17}$	$3.45\text{x}10^{-7}$
	L	$3.44\text{x}10^{-17}$	$9.48\text{x}10^{-7}$	$5.16\text{x}10^{-17}$	$3.57\text{x}10^{-7}$	$9.21\text{x}10^{-17}$	$7.38\text{x}10^{-7}$
100	K	$3.95\text{x}10^{-17}$	$1.56\text{x}10^{-7}$	$8.12\text{x}10^{-17}$	$9.63\text{x}10^{-8}$	$8.60\text{x}10^{-17}$	$8.67\text{x}10^{-8}$
	L	$4.86\text{x}10^{-17}$	$2.43\text{x}10^{-7}$	$7.38\text{x}10^{-17}$	$8.94\text{x}10^{-8}$	$8.24\text{x}10^{-17}$	$1.87\text{x}10^{-7}$
200	K	$5.11\text{x}10^{-17}$	$3.88\text{x}10^{-8}$	$8.69\text{x}10^{-17}$	$2.62\text{x}10^{-8}$	$1.04\text{x}10^{-16}$	$2.69\text{x}10^{-8}$
	L	$5.85\text{x}10^{-17}$	$6.13\text{x}10^{-8}$	$7.06\text{x}10^{-17}$	$2.32\text{x}10^{-8}$	$9.49\text{x}10^{-17}$	$5.03\text{x}10^{-8}$

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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
No. of cells in each pipe	L1-error for variable	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	$4.69\text{x}10^{-17}$	$2.53\text{x}10^{-8}$	$7.45\text{x}10^{-17}$	$2.32\text{x}10^{-8}$	$5.76\text{x}10^{-17}$	$2.59\text{x}10^{-8}$

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