In ([10], JCP 227 No. 6, 2008, pp. 3101–3211), the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical ...

We present a new smoothness indicator that evaluates the local smoothness of a function inside of a stencil. The corresponding weighted essentially non-oscillatory (WENO) finite difference scheme can provide the fifth convergence order in smooth regions. The proposed WENO scheme provides at least the same or improved behavior over the classical fifth-order WENO scheme [9] and other fifth-order ...

Weighted essentially non-oscillatory (WENO) schemes are effective numerical methods to compute flows having shocks and steep gradients. WENO schemes are based on the successful essentially non-oscillatory (ENO) schemes introduced in [9] and [10]. The initial WENO scheme is constructed in [6], and further research in WENO can be found in papers such as [2], [4] and [5]. Both ENO and WENO schemes...

Polyharmonic splines are utilized in the WENO reconstruction of finite volume discretizations, yielding a numerical method for scalar conservation laws of arbitrary high order. The resulting WENO reconstruction method is, unlike previous WENO schemes using polynomial reconstructions, numerically stable and very flexible. Moreover, due to the theory of polyharmonic splines, optimal reconstructio...

In this paper, three versions of WENO schemes WENO-JS (JCP 126, 1996), WENO-M (JCP 207, 2005) and WENO-Z (JCP 227, 2008) are used for one-dimensional detonation wave simulations with fifth order characteristic based spatial flux reconstruction. Numerical schemes for solving the system of hyperbolic conversation laws using the ZND analytical solution as initial condition are presented. Numerical...

The purpose of this paper is twofold. Firstly we carry out an extension of the finite-volume WENO approach to three space dimensions and higher orders of spatial accuracy (up to eleventh order). Secondly, we propose to use three new fluxes as a building block in WENO schemes. These are the one-stage HLLC [29] and FORCE [24] fluxes and a recent multistage MUSTA flux [26]. The numerical results i...

In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) nite diierence schemes of Liu, Osher and Chan 9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L 1 norm) ENO schemes are (r +1) th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of...

Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient. The first advance consists of realizing that WENO schemes require us to carry out stencil operations very efficiently. In this pape...

We will present our recent result on the construction of high order WENO finite volume methods for the approximation of hyperbolic partial differential equations on Cartesian grids. The simplest way to use WENO methods on multidimensional Cartesian grids consists in applying a one-dimensional WENO scheme in each direction. This spatial discretization is typically combined with a Runge-Kutta met...