It is critical for a numerical scheme to obtain numerical results as accurate as possible with limited computational resources. Turbulent processes are very sensitive to numerical dissipation, which may dissipate the small length scales. On the other hand, dealing with shock waves, capturing and reproducing of the discontinuity may lead to non-physical oscillations for non-dissipative schemes. ...

A quantitative study is carried out in this paper to investigate the size of numerical viscosities and the resolution power of high-order weighted essentially nonoscillatory (WENO) schemes for solving one- and two-dimensional Navier-Stokes equations for compressible gas dynamics with high Reynolds numbers. A one-dimensional shock tube problem, a one-dimensional example with parameters motivated...

We develop an alternative formulation of conservative finite difference weighted essentially non-oscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive...

Stratocumulus clouds are the most common type of boundary layer cloud; their radiative effects strongly modulate climate. Large eddy simulations (LES) of stratocumulus clouds often struggle to maintain fidelity to observations because of the sharp gradients occurring at the entrainment interfacial layer at the cloud top. The challenge posed to LES by stratocumulus clouds is evident in the wide ...

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the c...

In this paper we present a high order weighted essentially non-oscillatory (WENO) scheme for solving a multi-class extension of the Lighthill-Whitham-Richards (LWR) model. We first review the multi-class LWR model and present some of its analytical properties. We then present the WENO schemes, which were originally designed for computational fluid dynamics problems and for solving hyperbolic co...

We present a new, formally third order, implicit Weighted Essentially NonOscillatory (iWENO3) finite volume scheme for solving systems of nonlinear conservation laws. We then generalize it to define an implicit Eulerian-Lagrangian WENO (iEL-WENO) scheme. Implicitness comes from the use of an implicit Runge-Kutta (RK) time integrator. A specially chosen two-stage RK method allows us to drastical...

Two-dimensional simulations of the single-mode Richtmyer–Meshkov instability (RMI) are conducted and compared to experimental results of Jacobs and Krivets (2005 Phys. Fluids 17 034105). The employed adaptive central-upwind sixth-order weighted essentially non-oscillatory (WENO) scheme (Hu X Y et al 2010 J. Comput. Phys. 229 8952–65) introduces only very small numerical dissipation while preser...

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation t...

An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline WENO reconstruction with high order flux evaluation. Polyharmonic splines are utilised in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible and optimal in the associated Sobolev (BeppoLevi) space. The flux evaluation is ...