We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus o...

In this paper, we review and construct fifthand ninth-order central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear weights appear in such a formulation and they are treated using the technique recently introduced by ...

We present a newly developed cosmological hydrodynamics code based on weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. WENO is a higher order accurate finite difference scheme designed for problems with piecewise smooth solutions containing discontinuities, and has been successful in application for problems involving bo...

Based on the Weighted Essential Non-Oscillatory (WENO) reconstruction for the macroscopic flow variables and the direct use of the corresponding gas distribution function of the NavierStokes (NS) solution, a high-order finite volume gas-kinetic scheme is constructed. Different from the previous high-order gas-kinetic method [Li, Xu, and Fu, A high-order gas-kinetic Navier-Stokes solver, J. Comp...

In this paper we develop high order positivity-preserving finite volume weighted essentially non-oscillatory (WENO) schemes for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. We carefully treat the technical complications in boundary conditions and global integration terms to ensure high order accuracy and positivity-preserving p...

A large-eddy simulation (LES) framework is developed for simulating the dynamics of clouds and boundary layers with closed water and entropy balances. The framework is based on the anelastic equations in a formulation that remains accurate for deep convection. As prognostic variables, it uses total water and entropy, which are conserved in adiabatic and reversible processes, including reversibl...

This paper investigates the accuracy and robustness of high order WENO schemes for predicting hypersonic shock wave/boundary layer interaction. The implicit time marching method with unfactored Gauss-Seidel line relaxation is used with a 5th order WENO finite difference scheme for inviscid fluxes. The viscous terms are discretized using a 4th order conservative central differencing. Numerical r...

A set of conservative 4th-order central differencing schemes for the viscous terms of Navier-Stokes equations are proved in this paper. These schemes are used with the 5th order WENO schemes for inviscid flux. The stencil width of the central differencing scheme is within that of the WENO scheme. The algorithm is used to simulate the vortex-induced oscillations of an elastically mounted circula...

We present two versions of third order accurate jet schemes, which achieve high order accuracy by tracking derivative information of the solution along characteristic curves. For a benchmark linear advection problem, the efficiency of jet schemes is compared with WENO and Discontinuous Galerkin methods of the same order. It is demonstrated that jet schemes possess the simplicity and speed of WE...

In this paper we construct high-order weighted essentially nonoscillatory (WENO) schemes for solving the nonlinear Hamilton–Jacobi equations on two-dimensional unstructured meshes. The main ideas are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy ...