Abstract
In this paper, we present a positivity-preserving high order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations based on the framework for constructing uniformly high order accurate positivity-preserving discontinuous Galerkin and finite volume schemes for Euler equations proposed in Zhang and Shu (J Comput Phys 230:1238–1248, 2011). The major advantages of the HWENO schemes is their compactness in the spacial field because the function and its first derivative are evolved in time and used in the reconstructions. On the other hand, the HWENO reconstruction tends to be more oscillatory than those of conventional WENO schemes. Thus positivity preserving techniques are more needed in HWENO schemes for the sake of stability. Numerical tests will be shown to demonstrate the robustness and high-resolution of the schemes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Capdeville, G.: A Hermite upwind WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 227, 2430–2454 (2008)
Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin method for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)
Ha, Y., Gardner, C.L.: Positive scheme numerical simulation of high mach number astrophysical jets. J. Sci. Comput. 34, 247–259 (2008)
Ha, Y., Gardner, C., Gelb, A., Shu, C.-W.: Numerical simulation of high Mach number astrophysical jets with radiative cooling. J. Sci. Comput. 24, 597–612 (2005)
Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169–180 (2013)
Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Korobeinikov, V.P.: Problems of Point-Blast Theory. American Institute of Physics, College Park (1991)
Lele, S.K.: Compact finite-difference schemes with spectra-like resolution. J. Comput. Phys. 103, 16–42 (1992)
Liu, H., Qiu, J.: Finite difference Hermite WENO schemes for conservation laws. J. Sci. Comput. 63, 548–572 (2015)
Perthame, B., Shu, C.-W.: On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73, 119–130 (1996)
Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 193, 115–135 (2004)
Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method II: Two dimensional case. Comput. Fluids 34, 642–663 (2005)
Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Van-Leer, B.: Towards the ultimate conservative difference scheme: III. A new approach to numerical convection. J. Comput. Phys. 23, 276–299 (1977)
Wang, C., Zhang, X., Shu, C.-W., Ning, J.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231, 653–665 (2012)
Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33, 1476–1493 (2010)
Xing, Y., Zhang, X.: Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput. 57, 19–41 (2013)
Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)
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)
Zhang, X., Shu, C.-W.: Positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comput. Phys. 230, 1238–1248 (2011)
Zhang, X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high order schemes for conservation laws: survey and new developments. Proc. R. Soc. A 467, 2752–2776 (2011)
Zhang, X., Xia, Y., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. J. Sci. Comput. 50, 29–62 (2012)
Zhang, X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012)
Zhang, X., Liu, Y.-Y., Shu, C.-W.: Maximum-principle-satisfying high order finite volume WENO schemes for convection–diffusion equations. SIAM J. Sci. Comput. 34, A627–A658 (2012)
Zhang, Y., Zhang, X., Shu, C.-W.: Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes. J. Comput. Phys. 234, 295–316 (2013)
Zhang, X., Shu, C.-W.: A minimum entropy principle of high order schemes for gas dynamics equations. Numer. Math. 121, 545–563 (2012)
Zhu, J., Qiu, J.: A class of fourth order finite volume hermite weighted essentially non-oscillatory schemes. Sci. China Ser. A Math. 51, 1549–1560 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research was supported by NSFC Grants 91230110, 11328104, 11571290 and the NSF Grant DMS-1522593.
Rights and permissions
About this article
Cite this article
Cai, X., Zhang, X. & Qiu, J. Positivity-Preserving High Order Finite Volume HWENO Schemes for Compressible Euler Equations. J Sci Comput 68, 464–483 (2016). https://doi.org/10.1007/s10915-015-0147-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-015-0147-8