High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

doi:10.1007/s42967-020-00117-y

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (1): 31-63.doi: 10.1007/s42967-020-00117-y

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High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

Jie Du^1,3, Yang Yang²

1 Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China;
2 Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA;
3 Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China

Received:2020-09-11 Revised:2020-12-18 Online:2023-03-20 Published:2023-03-08
Contact: Yang Yang,E-mail:yyang7@mtu.edu;Jie Du,E-mail:jdu@tsinghua.edu.cn E-mail:yyang7@mtu.edu;jdu@tsinghua.edu.cn
Supported by:
Jie Du is supported by the National Natural Science Foundation of China under Grant Number NSFC 11801302 and Tsinghua University Initiative Scientific Research Program. Yang Yang is supported by the NSF Grant DMS-1818467.

Abstract

Abstract: In this paper, we apply high-order finite difference (FD) schemes for multispecies and multireaction detonations (MMD). In MMD, the density and pressure are positive and the mass fraction of the ith species in the chemical reaction, say z_i, is between 0 and 1, with Σz_i = 1. Due to the lack of maximum-principle, most of the previous bound-preserving technique cannot be applied directly. To preserve those bounds, we will use the positivity-preserving technique to all the z_i'is and enforce Σz_i = 1 by constructing conservative schemes, thanks to conservative time integrations and consistent numerical fluxes in the system. Moreover, detonation is an extreme singular mode of flame propagation in premixed gas, and the model contains a significant stiff source. It is well known that for hyperbolic equations with stiff source, the transition points in the numerical approximations near the shocks may trigger spurious shock speed, leading to wrong shock position. Intuitively, the high-order weighted essentially non-oscillatory (WENO) scheme, which can suppress oscillations near the discontinuities, would be a good choice for spatial discretization. However, with the nonlinear weights, the numerical fluxes are no longer “consistent”, leading to nonconservative numerical schemes and the bound-preserving technique does not work. Numerical experiments demonstrate that, without further numerical techniques such as subcell resolutions, the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.

Key words: Weighted essentially non-oscillatory scheme, Finite difference method, Stiff source, Detonations, Bound-preserving, Conservative

CLC Number:

65M06
65M12

Jie Du, Yang Yang. High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 31-63.

TrendMD

References

1. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
2. Bao, W., Jin, S.: The random projection method for stiff detonation capturing. SIAM J. Sci. Comput. 23, 1000–1025 (2001)
3. Bao, W., Jin, S.: The random projection method for stiff multispecies detonation capturing. J. Comput. Phys. 178, 37–57 (2002)
4. Bihari, B., Schwendeman, D.: Multiresolution schemes for the reactive Euler equations. J. Comput. Phys. 154, 197–230 (1999)
5. Christlieb, A., Liu, Y., Tang, Q., Xu, Z.: High order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes. J. Comput. Phys. 281, 334–351 (2015)
6. Chuenjarern, N., Xu, Z., Yang, Y.: High-order bound-preserving discontinuous Galerkin methods for compressible miscible displacements in porous media on triangular meshes. J. Comput. Phys. 378, 110–128 (2019)
7. Clarke, J.F., Karni, S., Quirk, J.J., Roe, P.L., Simmonds, L.G., Toro, E.F.: Numerical computation of two-dimensional unsteady detonation waves in high energy solids. J. Comput. Phys. 106, 215–233 (1993)
8. Du, J., Wang, C., Qian, C., Yang, Y.: High-order bound-preserving discontinuous Galerkin methods for stiff multispecies detonation. SIAM J. Sci. Comput. 41, B250–B273 (2019)
9. Du, J., Yang, Y.: Third-order conservative sign-preserving and steady-state-preserving time integrations and applications in stiff multispecies and multireaction detonations. J. Comput. Phys. 395, 489–510 (2019)
10. Guo, H., Yang, Y.: Bound-preserving discontinuous Galerkin method for compressible miscible displacement problem in porous media. SIAM J. Sci. Comput. 39, A1969–A1990 (2017)
11. Guo, H., Liu, X., Yang, Y.: High-order bound-preserving finite difference methods for miscible displacements in porous media. J. Comput. Phys. 406(24), 109219 (2020)
12. Huang, J., Shu, C.-W.: Bound-preserving modified exponential Runge-Kutta discontinuous Galerkin methods for scalar hyperbolic equations with stiff source terms. J. Comput. Phys. 361, 111–135 (2018)
13. Huang, J., Shu, C.-W.: Positivity-preserving time discretizations for production-destruction equations with applications to non-equilibrium flows. J. Sci. Comput. 78, 1811–1839 (2019)
14. Huang, J., Zhao, W., Shu, C.-W.: A third-order unconditionally positivity-preserving scheme for production-destruction equations with applications to non-equilibrium flows. J. Sci. Comput. 79, 1015–1056 (2019)
15. Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
16. Kopecz, S., Meister, A.: On order conditions for modified Patankar-Runge-Kutta schemes. Appl. Numer. Math. 123, 159–179 (2018)
17. Kopecz, S., Meister, A.: Unconditionally positive and conservative third order modified PatankarRunge-Kutta discretizations of production-destruction systems. BIT Numer. Math. 58, 691–728 (2018)
18. LeVeque, R.J., Yee, H.C.: A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86, 187–210 (1990)
19. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
20. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697. Springer, Berlin, Heidelberg (1998)
21. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
22. Sun, Y., Engquist, B.: Heterogeneous multiscale methods for interface tracking of combustion fronts. Multiscale Model. Simul. 5, 532–563 (2006)
23. Wang, W., Shu, C.-W., Yee, H.C., Kotov, D.V., Sjögreen, B.: High order finite difference methods with subcell resolution for stiff multispecies detonation capturing. Commun. Comput. Phys. 17, 317–336 (2015)
24. Xiong, T., Qiu, J.-M., Xu, Z.: Parametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations. J. Sci. Comput. 67, 1066-1088 (2016)
25. Xu, Z.: Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one dimensional scalar problem. Math. Comput. 83, 2213–2238 (2014)
26. Xu, Z., Yang, Y., Guo, H.: High-order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes. J. Comput. Phys. 390, 323–341 (2019)
27. Yee, H.C., Kotov, D.V., Wang, W., Shu, C.-W.: Spurious behavior of shock-capturing methods by the fractional step approach: problems containing stiff source terms and discontinuities. J. Comput. Phys. 241, 266–291 (2013)
28. Zhang, X., Shu, C.-W.: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

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High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

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