In this paper, a speed-up strategy for finite volume WENO schemes is developed for solving hyperbolic conservation laws. It adopts p-adaptive like reconstruction, which automatically adjusts from fifth order WENO reconstruction to first order constant reconstruction when nearly constant solutions are detected by the undivided differences. The corresponding order of accuracy for the solutions is...

In this paper, a class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one-dimensional nonlinear hyperbolic conservation law systems is presented. The construction of HWENO schemes is based on a finite volume formulation, Hermite interpolation, and nonlinearly stable Runge–Kutta methods. The idea o...

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving Hamilton–Jacobi equations is presented. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, ...

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with d...

In this paper, we apply the high order WENO schemes to uniform cylindrical and spherical grid. Many 2-D and 3-D problems can be solved in 1-D equations if they have angular and radial symmetry. The reduced equations will typically involve geometric source terms. Therefore, conventional numerical schemes for Cartesian grid may not work well. We propose several approaches to apply the high order ...

We develop in this article an improved version of the fifth-order weighted essentially non-oscillatory (WENO) scheme. Through the novel use of higher order information already present in the framework of the classical scheme, new smoothness indicators are devised and we obtain a new WENO scheme with less dissipation than the classical WENO of Jiang and Shu [2], with the same computational cost,...

Keywords: Adaptive mesh refinement (AMR) WENO High order finite difference Multiscale simulations a b s t r a c t In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework [4,5] with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the t...

In this paper, an improved seventh-order WENO (WENO-Z7) scheme is suggested by extending the 5th-order WENO scheme of Borges et al[R. Borges, M. Carmona, B. Costa, W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227(2008) 3191-3211]. The sufficient condition for seventh-order accuracy is described for the new smoothness indic...

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

We introduce a multi-dimensional point-wise multi-domain hybrid Fourier-Continuation/WENO technique (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and essentially dispersionless, spectral, solution away from discontinuities, as well as mild CFL constraints for explicit time stepping schemes. The hybrid scheme conjugates the expensive, s...