A modified fifth-order WENOZ method for hyperbolic conservation laws

Abstract

The paper analyses by Taylor series the several fifth-order of accuracy schemes for hyperbolic conservation laws: the classical WENOJS scheme [G.S. Jiang, and C.W.Shu. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126:202-228, 1996], the WENOM scheme [A.K. Henrick, T.D. Aslam, and J.M. powers. Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207:542-567, 2005], the WENOZ scheme [R. Borges, M. Carmona, B. Costa, and W.S. Don. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227:3191-3211, 2008] and the scheme, called WENOε here, [F. Aràndiga, A. Baeza, A.M. Belda, and P. Mulet. Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal., 49:893-915, 2011]. The order of weights of these four schemes agreed to the optimal weights is presented in detail. Then three prerequisites are developed if one intends to improve the WENOJS scheme: the scheme arrives the 5th-order at critical points; the weights of scheme approximate the optimal weights with high-order accuracy when solution is smooth; the scheme shouldn't introduce much oscillations intuitively in the vicinity of discontinuities. According to the prerequisites above, a new WENO scheme (MWENOZ) is devised which is similar to the WENOZ scheme. Finally, the method designed here is demonstrated robustly by applying it to 1D and 2D numerical simulations and its advantage compared with the WENOZ scheme seems more striking in 2D problems.

Numerical analysis of the mean structure of gaseous detonation with dilute water spray

“…A classical time-splitting method is employed to couple the hydrodynamics with the detailed chemistry and the interphase exchanges. The spatial derivatives of the left-hand side of (2.1)-(2.4) are discretized by fifth-order advection upstream splitting methods using pressure-based weight functions (known as AUSMPW+) improved by Kim, Kim & Rho (2001) based on a modified weighted essentially non-oscillatory scheme (known as MWENO-Z) (Hu, Wang & Chen 2016) and the diffusive terms by a second-order central differential scheme. The time integration method for the convective and diffusion terms is the third-order total variation diminishing Runge-Kutta method (Gottlieb, Shu & Tadmor 2001), and the multi-time-scale method (Gou et al 2010) is used for efficient time integration of the chemical source term.…”

An Efficient Mapped WENO Scheme Using Approximate Constant Mapping

“…Then, the weighting method presented in [27] and the smoothness indicators designed in [23] eventually became a standard, and the WENO-JS schemes especially the fifth-order one [23] developed into one of the most popular high-order methods [25]. In recent decades, many successful works have been done to raise some issues about WENO schemes [1,5,6,9,11,12,20,22,44].…”

An enhanced weighted essentially non-oscillatory high order scheme for explosion modelling

“…Since we were focused on physical explosion investigation, we consider Rayleigh-Taylor instability problem to verify the treatment of the source term. Similar analysis of the Rayleigh-Taylor instability problem can be found elsewhere (Borges et al 2008;Hu et al 2016).…”

“…It has been suggested that high-order numerical schemes emerge as a potential alternative for improvement of explosion studies. In the context of high-order schemes, WENO (weighted essentially non oscillatory) schemes have demonstrated to be robust and efficient (Borges et al 2008;Hu et al 2016;Acker et al 2016) and recent papers indicate that numerical modelling of explosions are under continuous development relying on WENO approaches (Wang et al 2013(Wang et al , 2015Xu et al 2018). Combined adaptive mesh refinement (AMR) to WENO scheme has been applied in the investigation of denotation (Wang et al 2015).…”