A high-order ERK-SUPG method with Said-Ball elements for modeling convection-dominated flows

Lanyin Sun; Ziwei Dong; Lanyin Sun; Ziwei Dong

doi:10.3934/math.20261044

AIMS Mathematics

2026, Volume 11, Issue 8: 26077-26096. doi: 10.3934/math.20261044

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A high-order ERK-SUPG method with Said-Ball elements for modeling convection-dominated flows

Lanyin Sun ^,,
Ziwei Dong

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, China

Received: 28 April 2026 Revised: 18 June 2026 Accepted: 28 July 2026 Published: 20 August 2026
MSC : 65D17, 65M60

This paper develops a high-order explicit Runge-Kutta streamline upwind/Petrov-Galerkin (ERK-SUPG) method to simulate convection-dominated flows in offshore hydrocarbon reservoir development. The method integrates explicit Runge-Kutta temporal discretization with a streamline upwind/Petrov-Galerkin spatial scheme based on Said-Ball basis functions, ensuring numerical stability and avoiding oscillations in high-order computations. Numerical results demonstrate that the ERK-SUPG method, with stabilization parameters dependent on time and spatial step sizes, achieve theoretical convergence rates of $ O(h^{2(q+1)/3}+\tau^p) $ and $ O(h^{q+1/2}+\tau^p) $, respectively. Numerical tests on reservoir-scale models, including scenarios with time-dependent convection and mixed boundary conditions, show errors reduced to $ 10^{-11} $ under mesh refinement, validating the method's robustness and effectiveness for subsea fluid migration analysis.
- offshore hydrocarbon reservoir development,
- convection-dominated equations,
- ERK method,
- SUPG method
Citation: Lanyin Sun, Ziwei Dong. A high-order ERK-SUPG method with Said-Ball elements for modeling convection-dominated flows[J]. AIMS Mathematics, 2026, 11(8): 26077-26096. doi: 10.3934/math.20261044

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Abstract

This paper develops a high-order explicit Runge-Kutta streamline upwind/Petrov-Galerkin (ERK-SUPG) method to simulate convection-dominated flows in offshore hydrocarbon reservoir development. The method integrates explicit Runge-Kutta temporal discretization with a streamline upwind/Petrov-Galerkin spatial scheme based on Said-Ball basis functions, ensuring numerical stability and avoiding oscillations in high-order computations. Numerical results demonstrate that the ERK-SUPG method, with stabilization parameters dependent on time and spatial step sizes, achieve theoretical convergence rates of $ O(h^{2(q+1)/3}+\tau^p) $ and $ O(h^{q+1/2}+\tau^p) $, respectively. Numerical tests on reservoir-scale models, including scenarios with time-dependent convection and mixed boundary conditions, show errors reduced to $ 10^{-11} $ under mesh refinement, validating the method's robustness and effectiveness for subsea fluid migration analysis.

References

[1]	L. Silva, C. Soares, Oilfield development system optimization under reservoir production uncertainty, Ocean Eng., 225 (2021), 108758. https://doi.org/10.1016/j.oceaneng.2021.108758 doi: 10.1016/j.oceaneng.2021.108758
[2]	J. Yang, The development and prospect of the offshore oil and gas industry, Advances in Resources Research, 1 (2021), 1–12. https://doi.org/https://doi.org/10.50908/arr.1.1_1 doi: 10.50908/arr.1.1_1
[3]	L. Silva, C. Soares, Optimisation of offshore oilfield development including the risk of failure of subsea manifolds, Ocean Eng., 324 (2025), 120668. https://doi.org/10.1016/j.oceaneng.2025.120668 doi: 10.1016/j.oceaneng.2025.120668
[4]	N. Babu, P. Lin, M. Abutaqiya, C. Sisco, J. Wang, F. Vargas, Systematic investigation of asphaltene deposition in the wellbore and near-wellbore region of a deepwater oil reservoir under gas injection, part 2: computational fluid dynamics modeling of asphaltene deposition, Energy Fuels, 33 (2019), 3645–3661. https://doi.org/10.1021/acs.energyfuels.8b03239 doi: 10.1021/acs.energyfuels.8b03239
[5]	B. Wu, B. Zhang, Z. Jin, Multi-mechanism coupled dynamic modeling and numerical simulation of microbial enhanced oil recovery in heterogeneous oil reservoirs, Advances in Resources Research, 5 (2025), 2601–2631. https://doi.org/10.50908/arr.5.4_2601 doi: 10.50908/arr.5.4_2601
[6]	A. Kim, Complete analytic solutions for convection-diffusion-reaction-source equations without using an inverse Laplace transform, Sci. Rep., 10 (2020), 8040. https://doi.org/10.1038/s41598-020-63982-w doi: 10.1038/s41598-020-63982-w
[7]	A. Askar, Á. Nagy, I. Barna, E. Kovács, Analytical and numerical results for the diffusion-reaction equation when the reaction coefficient depends on simultaneously the space and time coordinates, Computation, 11 (2023), 127. https://doi.org/10.3390/computation11070127 doi: 10.3390/computation11070127
[8]	T. Ludescher, S. Gross, A. Reusken, A multigrid method for unfitted finite element discretizations of elliptic interface problems, SIAM J. Sci. Comput., 42 (2020), A318–A342. https://doi.org/10.1137/18M1203353 doi: 10.1137/18M1203353
[9]	P. Theeraek, S. Phongthanapanich, P. Dechaumphai, Solving convection-diffusion-reaction equation by adaptive finite volume element method, Math. Comput. Simulat., 82 (2011), 220–233. https://doi.org/10.1016/j.matcom.2011.06.002 doi: 10.1016/j.matcom.2011.06.002
[10]	R. Lewis, A. Makurat, W. Pao, Fully coupled modeling of seabed subsidence and reservoir compaction of North Sea oil fields, Hydrogeol. J., 11 (2003), 142–161. https://doi.org/10.1007/s10040-002-0239-z doi: 10.1007/s10040-002-0239-z
[11]	M. Dunham, S. Ansari, C. Farquharson, Application of 3D marine controlled-source electromagnetic finite-element forward modeling to hydrocarbon exploration in the Flemish Pass Basin offshore Newfoundland, Canada, Geophysics, 83 (2018), WB33–WB49. https://doi.org/10.1190/geo2017-0451.1 doi: 10.1190/geo2017-0451.1
[12]	R. Allawi, M. Al-Jawad, 4D Finite element modeling of stress distribution in depleted reservoir of south Iraq oilfield, J. Petrol. Explor. Prod. Technol., 12 (2022), 679–700. https://doi.org/10.1007/s13202-021-01329-5 doi: 10.1007/s13202-021-01329-5
[13]	T. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal., 22 (1985), 970–1013. https://doi.org/10.1137/0722059 doi: 10.1137/0722059
[14]	E. do Carmo, G. Alvarez, A new stabilized finite element formulation for scalar convection-diffusion problems: the streamline and approximate upwind/Petrov-Galerkin method, Comput. Method. Appl. Mech., 192 (2003), 3379–3396. https://doi.org/10.1016/S0045-7825(03)00292-5 doi: 10.1016/S0045-7825(03)00292-5
[15]	E. Parish, M. Yano, I. Tezaur, T. Iliescu, Residual-based stabilized reduced-order models of the transient convection-diffusion-reaction equation obtained through discrete and continuous projection, Arch. Computat. Methods Eng., 32 (2025), 1885–1929. https://doi.org/10.1007/s11831-024-10197-1 doi: 10.1007/s11831-024-10197-1
[16]	A. Brooks, T. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible {N}avier-Stokes equations, Comput. Method. Appl. Mech., 32 (1982), 199–259. https://doi.org/10.1016/0045-7825(82)90071-8 doi: 10.1016/0045-7825(82)90071-8
[17]	T. Tezduyar, Y. Park, H. Deans, Finite element procedures for time‐dependent convection-diffusion-reaction systems, Int. J. Numer. Meth. Fl., 7 (1987), 1013–1033. https://doi.org/10.1002/fld.1650071003 doi: 10.1002/fld.1650071003
[18]	V. John, J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, SIAM J. Numer. Anal., 49 (2011), 1149–1176. https://doi.org/10.1137/100789002 doi: 10.1137/100789002
[19]	M. Esmaily, D. Jia, An augmented streamline upwind/Petrov-Galerkin method for the time-spectral convection-diffusion equation, J. Comput. Phys., 520 (2025), 113484. https://doi.org/10.1016/j.jcp.2024.113484 doi: 10.1016/j.jcp.2024.113484
[20]	C. Provatidis, Bézier versus Lagrange polynomials-based finite element analysis of 2-D potential problems, Adv. Eng. Softw., 73 (2014), 22–34. https://doi.org/10.1016/j.advengsoft.2014.02.006 doi: 10.1016/j.advengsoft.2014.02.006
[21]	M. Bacciocchi, N. Fantuzzi, Free vibrations and bending of laminated thin plates with distorted domains in gradient elasticity: an isoparametric finite element formulation based on Hermite mapping, Acta Mech., in press. https://doi.org/10.1007/s00707-025-04254-4
[22]	A. Rodríguez, L. Bruno, F. Rapetti, Whitney edge elements and the Runge phenomenon, J. Comput. Appl. Math., 427 (2023), 115117. https://doi.org/10.1016/j.cam.2023.115117 doi: 10.1016/j.cam.2023.115117
[23]	G. Hu, M. Li, J. Zhong, Combined cubic generalized Ball surfaces: construction and shape optimization using an enhanced JS algorithm, Adv. Eng. Softw., 176 (2023), 103404. https://doi.org/10.1016/j.advengsoft.2022.103404 doi: 10.1016/j.advengsoft.2022.103404
[24]	S. Zafar, M. Hussain, Fair curve designing by Said-Ball curve, PLoS One, 20 (2025), e0324553. https://doi.org/10.1371/journal.pone.0324553 doi: 10.1371/journal.pone.0324553
[25]	H. Wu, Dual bases for a new family of generalized Ball bases, J. Comput. Math., 22 (2004), 79–88.
[26]	L. Zhang, H. Wu, J. Tan, Dual bases for Wang-Bézier basis and their applications, Appl. Math. Comput., 214 (2009), 218–227. https://doi.org/10.1016/j.amc.2009.03.079 doi: 10.1016/j.amc.2009.03.079
[27]	G. Hu, W. Dou, B. Du, Degree reduction of ball Said-Ball surfaces using an enhanced coot optimization algorithm, Math. Method. Appl. Sci., 48 (2025), 8297–8321. https://doi.org/10.1002/mma.9791 doi: 10.1002/mma.9791
[28]	A. Stuart, A. Peplow, The dynamics of the theta method, SIAM Journal on Scientific and Statistical Computing, 12 (1991), 1351–1372. https://doi.org/10.1137/0912074 doi: 10.1137/0912074
[29]	B. Sanderse, B. Koren, Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations, J. Comput. Phys., 231 (2012), 3041–3063. https://doi.org/10.1016/j.jcp.2011.11.028 doi: 10.1016/j.jcp.2011.11.028
[30]	I. Faragó, Convergence and stability constant of the theta-method, Proceedings of Applications of Mathematics 2013, 2013, 42–51.
[31]	S. Keeling, Galerkin/Runge-Kutta discretizations for semilinear parabolic equations, SIAM J. Numer. Anal., 27 (1990), 394–418. https://doi.org/10.1137/0727024 doi: 10.1137/0727024
[32]	M. Song, Z. Yang, M. Liu, Stability of $\theta$-methods for advanced differential equations with piecewise continuous arguments, Comput. Math. Appl., 49 (2005), 1295–1301. https://doi.org/10.1016/j.camwa.2005.02.002 doi: 10.1016/j.camwa.2005.02.002
[33]	J. Butcher, A history of Runge-Kutta methods, Appl. Numer. Math., 20 (1996), 247–260. https://doi.org/10.1016/0168-9274(95)00108-5 doi: 10.1016/0168-9274(95)00108-5
[34]	Z. Sun, C. Shu, Strong stability of explicit Runge-Kutta time discretizations, SIAM J. Numer. Anal., 57 (2019), 1158–1182. https://doi.org/10.1137/18M122892X doi: 10.1137/18M122892X
[35]	C. Runge, Über die numerische Auflösung von differentialgleichungen, Math. Ann., 46 (1895), 167–178. https://doi.org/10.1007/BF01446807 doi: 10.1007/BF01446807
[36]	J. Butcher, Implicit Runge-Kutta processes, Math. Comp., 18 (1964), 50–64. https://doi.org/10.1090/S0025-5718-1964-0159424-9 doi: 10.1090/S0025-5718-1964-0159424-9
[37]	R. Alexander, Diagonally implicit Runge-Kutta methods for stiff ODE's, SIAM J. Numer. Anal., 14 (1977), 1006–1021. https://doi.org/10.1137/0714068 doi: 10.1137/0714068
[38]	G. Dimarco, L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., 51 (2013), 1064–1087. https://doi.org/10.1137/12087606X doi: 10.1137/12087606X
[39]	F. Ghoreishi, R. Ghaffari, N. Saad, Fractional order Runge-Kutta methods, Fractal Fract., 7 (2023), 245. https://doi.org/10.3390/fractalfract7030245 doi: 10.3390/fractalfract7030245
[40]	L. Sun, Z. Dong, A $\theta$-difference finite element method for solving convection-diffusion-reaction equations based on the Wang-Ball element, J. Appl. Math. Comput., 71 (2025), 7639–7673. https://doi.org/10.1007/s12190-025-02556-3 doi: 10.1007/s12190-025-02556-3
[41]	L. Zhang, J. Tan, Z. Dong, The dual bases for the Bézier-Said-Wang type generalized Ball polynomial bases and their applications, Appl. Math. Comput., 217 (2010), 3088–3101. https://doi.org/10.1016/j.amc.2010.08.041 doi: 10.1016/j.amc.2010.08.041
[42]	P. Ciarlet, The finite element method for elliptic problems, Philadelphia: Society for Industrial and Applied Mathematics, 2002. https://doi.org/10.1137/1.9780898719208
[43]	H. Roos, M. Stynes, L. Tobiska, Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, Berlin: Springer, 2008. https://doi.org/10.1007/978-3-540-34467-4
[44]	C. Lubich, A. Ostermann, Runge-Kutta approximation of quasi-linear parabolic equations, Math. Comp., 64 (1995), 601–627. https://doi.org/10.1090/S0025-5718-1995-1284670-0 doi: 10.1090/S0025-5718-1995-1284670-0
[45]	Q. Zhang, C. Shu, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal., 48 (2010), 1038–1063. https://doi.org/10.1137/090771363 doi: 10.1137/090771363

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