Accurate ENO-like schemes for the model of fluid flows in a nozzle with variable cross-section

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 10 خرداد 1404

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.87744.1431

نویسندگان

D.H. Cuong¹؛ M.D. Thanh^{* 2، 3}

¹Department of Mathematics, Ho Chi Minh City University of Education, 280 An Duong Vuong str., District 5, Ho Chi Minh City, Vietnam.

²Corresponding author, Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam.

³Vietnam National University, Ho Chi Minh City, Vietnam.

چکیده

A class of accurate high-order ENO-like schemes for the model of fluid flows in a nozzle with variable cross-section is presented. The model contains a nonconservative source term, which causes unsatisfactory results to standard numerical schemes, even for low-order ones. The proposed schemes rely on exact Riemann solvers and the reconstructed piecewise polynomials which are nonoscillatory. These schemes inherit the high-order precision of the ENO schemes like many existing ENO-type schemes, and possess a good accuracy. The ENO-like scheme corresponding to k = 3 can get the precision as good as van Leer-type schemes, and is numerically stable. Moreover, the ENO-like schemes for larger k may suffer from oscillations. 

کلیدواژه‌ها

Nonconservative؛ Numerical approximation؛ Essentially nonoscillatory scheme؛ Accuracy؛ Resonance

مراجع

[1] Ambroso, A., Chalons, C., Coquel, F. and Galié, T. Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, Math. Mod. Numer. Anal., 43 (2009), 1063–1097.
[2] Ambroso, A., Chalons, C. and Raviart, P.-A. A Godunov-type method for the seven-equation model of compressible two-phase flow, Computers & Fluids, 54 (2012), 67–91.
[3] Audusse, E., Bouchut, F., Bristeau, M-O., Klein, R. and Perthame, B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), 2050–2065.
[4] Baudin, M., Coquel, F. and Tran, Q.-H. A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline, SIAM J. Sci. Comput., 27 (2005), 914–936.
[5] Ben-Artzi, M. and Falcovitz, J. An upwind second-order scheme for compressible duct flows, SIAM J. Sci. and Stat. Comput., 7(2006), 744–768.

[6] Botchorishvili, R., Perthame, B. and Vasseur, A. Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comput., 72 (2003), 131–157.
[7] Botchorishvili, R. and Pironneau, O. Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws, J. Comput. Phys., 187 (2003), 391–427.
[8] Castro, M.J., Chalons, C., Del Grosso, A. and Morales de Luna, T. Lagrange-projection methods for shallow water equations with movable bottom and erosion-deposition processes Num. Math.: Theory, Meth. Appl., 16(2023), 1087–1126.
[9] Castro, C.E., and Toro, E.F. A Riemann solver and upwind methods for a two-phase flow model in non-conservative form, Internat. J. Numer. Methods Fluids, 50 (2006), 275–307.
[10] Chalons, C., Del Grosso, A.and Toro, E.F. Numerical approximation and uncertainty quantification for arterial blood flow models with viscoelasticity, J. Comput. Phys., 457 (2022), 111071.
[11] Chinnayya, A., LeRoux, A.-Y. and Seguin, N. A well-balanced numerical scheme for the approximation of the shallow water equations with topography: the resonance phenomenon, Int. J. Finite Vol., 1(4), 2004, 1–33.
[12] Coquel, F., El Amine, K., Godlewski, E., Perthame, B. and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows, J. Comput. Phys. 136 (1997), 272–288.
[13] Coquel, F., Hérard, J.-M., Saleh, K. and Seguin, N. Two properties of two-velocity two-pressure models for two-phase flows, Commun. Math. Sci. 12 (2014), 593–600.
[14] Coquel, F., Saleh, K. and Seguin, N. A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles, Math. Mod. Meth. Appl. Sci., 24 (2014), 2043–2083.

[15] Cuong, D.H. and Thanh, M.D. A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section, Appl. Math. Comput., 256 (2015) 602–629.
[16] Cuong, D.H. and Thanh, M.D. A high-resolution van Leer-type scheme for a model of fluid flows in a nozzle with variable cross-section, J. Korean Math. Soc., Vol. 54 (1) (2017), 141–175.
[17] Cuong, D.H. and Thanh, M.D. Computing algorithms in resonant regime for a two-phase flow model, Taiwan. J. Math. Dec. (2023), 1135–1168
[18] Dal Maso, G., LeFloch, P.G. and Murat, F. Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), 483–548.
[19] Gallouët, T., Hérard, J.-M. and Seguin, N. Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math. Models Methods Appl. Sci., 14 (2004) 663–700.
[20] Goatin, P. Macroscopic traffic flow modelling: from kinematic waves to autonomous vehicles, Commun. Appl. Ind. Math., 14(1) (2023), 1–16.
[21] Goatin, P. and LeFloch, P.G. The Riemann problem for a class of resonant nonlinear systems of balance laws, Ann. Inst. H. Poincare Anal. NonLineaire, 21 (2004), 881–902.
[22] Greenberg, J.M. and Leroux, A.Y. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996) 1–16.
[23] Han, X. and Li, G. Well-balanced finite difference WENO schemes for the Ripa model, Comput. Fluid. 134-135 (2016), 1–10.
[24] Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. Uniformly high order essentially non- oscillatory schemes, III, J. Comput. Phys., 71(1987), 231–303.
[25] Isaacson, E. and Temple, B. Nonlinear resonance in systems of conser-vation laws, SIAM J. Appl. Math., 52 (1992) 1260–1278.

[26] Isaacson, E. and Temple, B. Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995) 625–640.
[27] Kröner, D., LeFloch, P.G. and Thanh, M.D. The minimum entropy principle for fluid flows in a nozzle with discontinuous cross-section, Math. Mod. Numer. Anal., 42 (2008), 425–442.
[28] Kröner, D. and Thanh, M.D. Numerical solutions to compressible flows in a nozzle with variable cross-section, SIAM J. Numer. Anal., 43 (2005), 796–824.
[29] LeFloch, P.G. and Thanh, M.D. The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Comm. Math. Sci., 1 (2003), 763–797.
[30] LeFloch, P.G. and Thanh, M.D. A Godunov-type method for the shallow water equations with variable topography in the resonant regime, J. Comput. Phys., 230 (2011), 7631–7660.
[31] LeFloch, P.G. and Thanh, M.D. The Riemann problem in continuum physics, Appl. Math. Sci., Springer, 2024.
[32] Marchesin, D. and Paes-Leme, P.J. A Riemann problem in gas dynamics with bifurcation. Hyperbolic partial differential equations III, Comput. Math. Appl. (Part A), 12 (1986) 433–455.
[33] Munkejord, S.T. Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation, Computers & Fluids, 36 (2007), 1061–1080.
[34] Saurel, R. and Abgrall, R. A multi-phase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150 (1999), 425–467.
[35] Schwendeman, D.W., Wahle, C.W. and Kapila, A.K. The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J. Comput. Phys., 212 (2006), 490–526.

[36] Shen, H. A class of ENO schemes with adaptive order for solving hyperbolic conservation laws, Computers & Fluids, 266, (2023), 106050.
[37] Shu, C.W. Essentially non-oscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws. In: Quarteroni A. (eds) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol 1697, Springer, Berlin, Heidelberg (1998), 325–432.
[38] Thanh, M.D. The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math., 69 (2009), 1501–1519.
[39] Thanh, M.D., Kröner, D. and C. Chalons, A robust numerical method for approximating solutions of a model of two-phase flows and its properties, Appl. Math. Comput., 219 (2012), 320–344.
[40] Thanh, M.D., Kröner, D. and N.T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math., 61 (2011), 702–721.
[41] Thanh, N.X., Thanh, M.D. and Cuong, D.H. A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography, Adv. Comput. Math., 47 (2021) 1–53.
[42] Xu, C., Zhang, F., Dong, H. and Jiang, H. Arbitrary high-order extended essentially non-oscillatory schemes for hyperbolic conservation laws, Int. J. for Num. Meth. Fluids, 93(7) (2021), 2136–2154.