High-order finite-volume central targeted essentially non-oscillatory schemes for shock-driven flows on unstructured meshes

doi:10.1007/s10483-026-3381-7

Abstract

Abstract:

The high-order targeted essentially non-oscillatory (TENO) scheme, known for its innovative weighting strategy, has demonstrated strong potential for complex flow predictions and applications. This study extends the TENO weighting approach to develop a family of central TENO (CTENO) schemes for unstructured meshes. The CTENO schemes employ compact directional stencils, which increase the likelihood of finding all stencils within smooth regions. The design is intentionally compact to simplify the implementation of directional stencils. An effective scale-separation strategy is adopted via an essentially non-oscillatory (ENO)-like stencil selection method, which employs large central stencils in smooth areas to achieve high-order accuracy, and employs smaller directional stencils near discontinuities to improve shock-capturing capability. Extensive tests involving central weighted essentially non-oscillatory (CWENO), TENO, and CTENO family schemes are conducted to assess their performance in terms of accuracy, parallel scalability, and computational efficiency. The applications to shock-driven flows indicate that the proposed schemes deliver high-order accuracy, lower numerical dissipation, and excellent shock-capturing performance in several practical flow cases such as shock reflection, bubble dynamics, explosion, and particle flow problems.

Key words: high order, shock wave, unstructured mesh, compressible flow

2010 MSC Number:

O241

Qihang MA, Feng FENG, Bofu WANG, Quan ZHOU. High-order finite-volume central targeted essentially non-oscillatory schemes for shock-driven flows on unstructured meshes. Applied Mathematics and Mechanics (English Edition), 2026, 47(5): 1177-1204.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3381-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I5/1177

Figures/Tables 18

Fig. 1

Table 1

Errors and accuracy with 2D Euler equations for CTENO and CTENOZ schemes (h is the number of edges, Ne is the total number of elements, and both L∞ error and L2 error are calculated based on h and Ne, in all figures and tables throughout this paper, CTENO3 denotes the 3rd-order CTENO scheme. Likewise, CTENOZ3 and CTENOZ4 refer to the 3rd-order and 4th-order CTENOZ schemes, respectively, and other notations follow the same rule)"

	h	N_e	L^∞ error	L^∞ order	L² error	L² order
CTENO3	1/10	100	$1.06 × 10 − 1$		$7.54 × 10 − 2$
CTENOZ3	1/20	400	$1.97 × 10 − 2$	2.42	$1.40 × 10 − 2$	2.43
	1/40	1 600	$2.65 × 10 − 3$	2.90	$1.88 × 10 − 3$	2.90
	1/80	6 400	$3.35 × 10 − 4$	2.98	$2.37 × 10 − 4$	2.98
	1/160	25 600	$4.20 × 10 − 5$	3.00	$2.97 × 10 − 5$	3.00
CTENO4	1/10	100	$5.39 × 10 − 2$		$3.72 × 10 − 2$
CTENOZ4	1/20	400	$4.32 × 10 − 3$	3.64	$2.96 × 10 − 3$	3.65
	1/40	1 600	$2.80 × 10 − 4$	3.94	$1.95 × 10 − 4$	3.92
	1/80	6 400	$1.77 × 10 − 5$	3.99	$1.24 × 10 − 5$	3.97
	1/160	25 600	$1.11 × 10 − 6$	4.00	$7.80 × 10 − 7$	3.99
CTENO5	1/10	100	$2.31 × 10 − 2$		$1.63 × 10 − 2$
CTENOZ5	1/20	400	$8.69 × 10 − 4$	4.73	$6.18 × 10 − 4$	4.72
	1/40	1 600	$2.83 × 10 − 5$	4.94	$2.00 × 10 − 5$	4.95
	1/80	6 400	$8.93 × 10 − 7$	4.99	$6.32 × 10 − 7$	4.99
	1/160	25 600	$2.80 × 10 − 8$	4.99	$1.98 × 10 − 8$	4.99
CTENO6	1/10	100	$1.33 × 10 − 2$		$9.11 × 10 − 3$
CTENOZ6	1/20	400	$2.49 × 10 − 4$	5.73	$1.71 × 10 − 4$	5.73
	1/40	1 600	$3.90 × 10 − 6$	6.00	$2.73 × 10 − 6$	5.97
	1/80	6 400	$6.03 × 10 − 8$	6.01	$4.24 × 10 − 8$	6.01
	1/160	25 600	$9.42 × 10 − 10$	6.00	$6.61 × 10 − 10$	6.00
CTENO7	1/10	100	$9.51 × 10 − 3$		$6.63 × 10 − 3$
CTENOZ7	1/20	400	$6.87 × 10 − 5$	7.11	$4.87 × 10 − 5$	7.09
	1/40	1 600	$5.66 × 10 − 7$	6.92	$4.00 × 10 − 7$	6.93
	1/80	6 400	$4.93 × 10 − 9$	6.84	$3.48 × 10 − 9$	6.84
	1/160	25 600	$4.39 × 10 − 11$	6.81	$3.02 × 10 − 11$	6.85

Table 1

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Table 2

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References 78

[1]	JAMESON, A. Analysis and design of numerical schemes for gas dynamics, 1: artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. International Journal of Computational Fluid Dynamics, 4(3-4), 171–218 (1995)
[2]	HARTEN, A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 135(2), 260–278 (1983)
[3]	HARTEN, A. and OSHER, S. Uniformly high-order accurate nonoscillatory schemes. SIAM Journal on Numerical Analysis, 24(2), 279–309 (1987)
[4]	LIU, X. D., OSHER, S., and CHAN, T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 115(1), 200–212 (1994)
[5]	JIANG, G. S. and SHU, C. W. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1), 202–228 (1996)
[6]	HENRICK, A. K., ASLAM, T. D., and POWERS, J. M. Mapped weighted essentially nonoscillatory schemes: achieving optimal order near critical points. Journal of Computational Physics, 207(2), 542–567 (2005)
[7]	BORGES, R., CARMONA, M., COSTA, B., and DON, W. S. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics, 227(6), 3191–3211 (2008)
[8]	JOHNSEN, E., LARSSON, J., BHAGATWALA, A. V., CABOT, W. H., MOIN, P., OLSON, B. J., RAWAT, P. S., SHANKAR, S. K., SJÖGREEN, B., YEE, H. C., ZHONG, X. L., and LELE, S. K. Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. Journal of Computational Physics, 229(4), 1213–1237 (2010)
[9]	ACKER, F., BORGES, R. B. D. R., and COSTA, B. An improved WENO-Z scheme. Journal of Computational Physics, 313, 726–753 (2016)
[10]	HILL, D. J. and PULLIN, D. I. Hybrid tuned center-difference-weno method for large eddy simulations in the presence of strong shocks. Journal of Computational Physics, 194(2), 435–450 (2004)
[11]	WEIRS, V., CANDLER, G., WEIRS, V., and CANDLER, G. Optimization of weighted ENO schemes for DNS of compressible turbulence. 13th Computational Fluid Dynamics Conference, AIAA, Virginia, U.S.A. (1997)
[12]	YAN, Z. G., LIU, H. Y., MAO, M. L., ZHU, H. J., and DENG, X. G. New nonlinear weights for improving accuracy and resolution of weighted compact nonlinear scheme. Computers & Fluids, 127, 226–240 (2016)
[13]	SHU, C. W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numerica, 29, 701–762 (2020)
[14]	FU, L., HU, X. Y., and ADAMS, N. A. A family of high-order targeted eno schemes for compressiblefluid simulations. Journal of Computational Physics, 305, 333–359 (2016)
[15]	FU, L. A low-dissipation finite-volume method based on a new teno shock-capturing scheme. Computer Physics Communications, 235, 25–39 (2019)
[16]	YE, C. C., ZHANG, P. J. Y., WAN, Z. H., and SUN, D. J. An alternative formulation of targeted ENO scheme for hyperbolic conservation laws. Computers & Fluids, 238, 105368 (2022)
[17]	HAIMOVICH, O. and FRANKEL, S. H. Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method. Computers & Fluids, 146, 105–116 (2017)
[18]	DONG, H. B., FU, L., ZHANG, F., NULL, Y. L., and LIU, J. Detonation simulations with a fifth-order TENO scheme. Communications in Computational Physics, 25(5), 1357–1393 (2025)
[19]	MA, T. B., WANG, C. T., and XU, X. Z. Conservative high precision pseudo arc-length method for strong discontinuity of detonation wave. Applied Mathematics and Mechanics (English Edition), 43(3), 417–436 (2022) https://doi.org/10.1007/s10483-022-2817-9
[20]	LUSHER, D. J. and SANDHAM, N. D. Shock-wave/boundary-layer interactions in transitional rectangular duct flows. Flow, Turbulence and Combustion, 105(2), 649–670 (2020)
[21]	LUSHER, D. J. and SANDHAM, N. Assessment of low-dissipative shock-capturing schemes for transitional and turbulent shock interactions. AIAA Aviation 2019 Forum, AIAA, Virginia, U.S.A. (2019)
[22]	HE, K., LIU, H. W., GUO, T. B., LI, X. L., and HE, Z. W. A class of order-preserving, spectral-property-optimized, and shock-capturing finite difference schemes for compressible flows. Physics of Fluids, 37(2), 026110 (2025)
[23]	HOU, Y. H., JIN, K., FENG, Y. L., and ZHENG, X. J. High-order targeted essentially non-oscillatory scheme for two-fluid plasma model. Applied Mathematics and Mechanics (English Edition), 44(6), 941–960 (2023) https://doi.org/10.1007/s10483-023-3003-6
[24]	FU, L. Review of the high-order TENO schemes for compressible gas dynamics and turbulence. Archives of Computational Methods in Engineering, 30(4), 2493–2526 (2023)
[25]	JI, Z., LIANG, T., and FU, L. High-order finite-volume TENO schemes with dual ENO-like stencil selection for unstructured meshes. Journal of Scientific Computing, 95(3), 76 (2023)
[26]	VINCENT, P. E. and JAMESON, A. Facilitating the adoption of unstructured high-order methods amongst a wider community of fluid dynamicists. Mathematical Modelling of Natural Phenomena, 6(3), 97–140 (2011)
[27]	LI, Z. J., SHEN, Y., and DING, H. Flow pattern and wave propagation induced by local energy deposition at droplet surface. Science China Physics, Mechanics & Astronomy, 68(9), 294706 (2025)
[28]	HU, Y., HUANG, Z. L., CAO, Y., SUN, Q. H., HUANG, H. J., DING, C. Y., and WANG, Z. Exploration of electron and ion thermodynamic models in laser-plasma expansion. Science China Physics, Mechanics & Astronomy, 68(9), 294707 (2025)
[29]	MA, Q. H., CHONG, K., WANG, B. F., and ZHOU, Q. Strong shock propagation for the finite-source circular blast in a confined domain. Applied Mathematics and Mechanics (English Edition), 45(6), 1071–1084 (2024) https://doi.org/10.1007/s10483-024-3120-7
[30]	MA, Q. H., WANG, B. F., and ZHOU, Q. Asymptotic self-similar solution for a finite-source spherical blast wave in power-law density media. Applied Mathematics and Mechanics (English Edition), 46(9), 1771–1786 (2025) https://doi.org/10.1007/s10483-025-3288-7
[31]	HU, C. Q. and SHU, C. W. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics, 150(1), 97–127 (1999)
[32]	ZHANG, Y. T. and SHU, C. W. Third order weno scheme on three dimensional tetrahedral meshes. Communications in Computational Physics, 5(2), 836–848 (2009)
[33]	SHI, J., HU, C. Q., and SHU, C. W. A technique of treating negative weights in WENO schemes. Journal of Computational Physics, 175(1), 108–127 (2002)
[34]	ZHU, J. and SHU, C. W. A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes. Journal of Computational Physics, 392, 19–33 (2019)
[35]	BALSARA, D. S., GARAIN, S., FLORINSKI, V., and BOSCHERI, W. An efficient class of WENO schemes with adaptive order for unstructured meshes. Journal of Computational Physics, 404, 109062 (2020)
[36]	DUMBSER, M. and KÄSER, M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics, 221(2), 693–723 (2007)
[37]	LEVY, D., PUPPO, G., and RUSSO, G. Compact central WENO schemes for multidimensional conservation laws. SIAM Journal on Scientific Computing, 22(2), 656–672 (2000)
[38]	DUMBSER, M., BOSCHERI, W., SEMPLICE, M., and RUSSO, G. Central weighted ENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. SIAM Journal on Scientific Computing, 39(6), A2564–A2591 (2017)
[39]	TSOUTSANIS, P. and DUMBSER, M. Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes. Computers & Fluids, 225, 104961 (2021)
[40]	TSOUTSANIS, P., ADEBAYO, E. M., MERINO, A. C., ARJONA, A. P., and SKOTE, M. CWENO finite-volume interface capturing schemes for multicomponent flows using unstructured meshes. Journal of Scientific Computing, 89(3), 64 (2021)
[41]	JI, Z., LIANG, T., and FU, L. A class of new high-order finite-volume TENO schemes for hyperbolic conservation laws with unstructured meshes. Journal of Scientific Computing, 92(2), 61 (2022)
[42]	TSOUTSANIS, P. Stencil selection algorithms for WENO schemes on unstructured meshes. Journal of Computational Physics, 475, 108840 (2023)
[43]	ANTONIADIS, A. F., DRIKAKIS, D., FARMAKIS, P. S., FU, L., KOKKINAKIS, I., NOGUEIRA, X., SILVA, P. A. S. F., SKOTE, M., TITAREV, V., and TSOUTSANIS, P. UCNS3D: an open-source high-order finite-volume unstructured CFD solver. Computer Physics Communications, 279, 108453 (2022)
[44]	KARYPIS, G. and KUMAR, V. Multilevelk-way partitioning scheme for irregular graphs. Journal of Parallel and Distributed Computing, 48(1), 96–129 (1998)
[45]	TSOUTSANIS, P., ANTONIADIS, A. F., and JENKINS, K. W. Improvement of the computational performance of a parallel unstructured WENO finite volume CFD code for Implicit Large Eddy Simulation. Computers & Fluids, 173, 157–170 (2018)
[46]	MENG, X. H., WU, J. Z., CHONG, K. L., and ZHOU, Q. Thermal and kinetic energy dissipation rates in vertical convection over rough sidewalls. Physics of Fluids, 38(3), 035137 (2026)
[47]	MENG, W. S., ZHAO, C. B., WU, J. Z., WANG, B. F., ZHOU, Q., and CHONG, K. L. Simulation of flow and debris migration in extreme ultraviolet source vessel. Physics of Fluids, 36(2), 023322 (2024)
[48]	STROUD, A. Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., New Jersey (1971)
[49]	STROUD, A. H. Some fourth degree integration formulas for simplexes. Mathematics of Computation, 30(134), 291–294 (1976)
[50]	GOTTLIEB, S. and SHU, C. W. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 67(221), 73–85 (1998)
[51]	TSOUTSANIS, P., ANTONIADIS, A. F., and DRIKAKIS, D. WENO schemes on arbitrary unstructured meshes for laminar, transitional and turbulent flows. Journal of Computational Physics, 256(C), 254–276 (2014)
[52]	STEWART, G. W. Matrix Algorithms, Volume 1: Basic Decompositions, Society for Industrial and Applied Mathematics, Philadelphia, U.S.A. (1998)
[53]	FU, L. A very-high-order TENO scheme for all-speed gas dynamics and turbulence. Computer Physics Communications, 244, 117–131 (2019)
[54]	FU, L., HU, X. Y., and ADAMS, N. A. A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws. Journal of Computational Physics, 374, 724–751 (2018)
[55]	FU, L., HU, X. Y., and ADAMS, N. A. Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws. Journal of Computational Physics, 349, 97–121 (2017)
[56]	CASTRO, M., COSTA, B., and DON, W. S. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. Journal of Computational Physics, 230(5), 1766–1792 (2011)
[57]	TORO, E. F., SPRUCE, M., and SPEARES, W. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4(1), 25–34 (1994)
[58]	SHU, C. W. and OSHER, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 77(2), 439–471 (1988)
[59]	WOODWARD, P. and COLELLA, P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 54(1), 115–173 (1984)
[60]	TSOUTSANIS, P. Extended bounds limiter for high-order finite-volume schemes on unstructured meshes. Journal of Computational Physics, 362(C), 69–94 (2018)
[61]	MILTON, V. D. and FRANK, M. W. An album of fluid motion. Journal of Fluid Engineering, 104, 542–543 (1982)
[62]	DUMBSER, M., KÄSER, M., TITAREV, V. A., and TORO, E. F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. Journal of Computational Physics, 226(1), 204–243 (2007)
[63]	GUAN, H., WANG, J. C., WEI, Z. J., and WU, C. J. Numerical analysis of the interaction of 3D compressible bubble clusters. Applied Mathematics and Mechanics (English Edition), 40(8), 1181–1196 (2019) https://doi.org/10.1007/s10483-019-2509-6
[64]	ALLAIRE, G., CLERC, S., and KOKH, S. A five-equation model for the simulation of interfaces between compressible fluids. Journal of Computational Physics, 181(2), 577–616 (2002)
[65]	HAAS, J. F. and STURTEVANT, B. Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. Journal of Fluid Mechanics, 181, 41–76 (1987)
[66]	CORALIC, V. and COLONIUS, T. Finite-volume WENO scheme for viscous compressible multicomponent flows. Journal of Computational Physics, 274(C), 95–121 (2014)
[67]	TERASHIMA, H. and TRYGGVASON, G. A front-tracking/ghost-fluid method for fluid interfaces in compressible flows. Journal of Computational Physics, 228(11), 4012–4037 (2009)
[68]	QUIRK, J. J. and KARNI, S. On the dynamics of a shock-bubble interaction. Journal of Fluid Mechanics, 318, 129–163 (1996)
[69]	BRODE, H. L. Numerical solutions of spherical blast waves. Journal of Applied Physics, 26(6), 766–775 (1955)
[70]	HARILAL, S. S., BINDHU, C. V., TILLACK, M. S., NAJMABADI, F., and GAERIS, A. C. Internal structure and expansion dynamics of laser ablation plumes into ambient gases. Journal of Applied Physics, 93(5), 2380–2388 (2003)
[71]	ZHAO, Z. Y. and LI, W. Z. Numerical simulation of laser-produced plasma expansion on a droplet surface. Scientific Reports, 13, 4085 (2023)
[72]	PRICE, M. A., NGUYEN, V. T., HASSAN, O., and MORGAN, K. An Euler-Lagrange particle approach for modeling fragments accelerated by explosive detonation. International Journal for Numerical Methods in Engineering, 106(11), 904–926 (2016)
[73]	CAPECELATRO, J. and WAGNER, J. L. Gas-particle dynamics in high-speed flows. Annual Review of Fluid Mechanics, 56, 379–403 (2024)
[74]	PARMAR, M., HASELBACHER, A., and BALACHANDAR, S. On the unsteady inviscid force on cylinders and spheres in subcritical compressible flow. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366(1873), 2161–2175 (2008)
[75]	LING, Y. and BALACHANDAR, S. Simulation and scaling analysis of a spherical particle-laden blast wave. Shock Waves, 28(3), 545–558 (2018)
[76]	JOURDAN, G., HOUAS, L., IGRA, O., ESTIVALEZES, J. L., DEVALS, C., and MESHKOV, E. E. Drag coefficient of a sphere in a non-stationary flow: new results. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2088), 3323–3345 (2007)
[77]	BALAKRISHNAN, K., NANCE, D. V., and MENON, S. Simulation of impulse effects from explosive charges containing metal particles. Shock Waves, 20(3), 217–239 (2010)
[78]	WAGNER, J. L., BERESH, S. J., KEARNEY, S. P., PRUETT, B. O. M., and WRIGHT, E. K. Shock tube investigation of quasi-steady drag in shock-particle interactions. Physics of Fluids, 24(12), 123301 (2012)

Element	CWENO5	TENO5	CTENO5	CTENOZ5
369 604	1.000	1.036	1.033	1.000
714 138	2.798	2.870	2.795	2.811