High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

Electrification and lightning are simulated for a small continental multicell storm. The results are consistent with observations and thus provide additional understanding of the charging processes and evolution of this storm. The first six observed lightning flashes were all negative cloud-to-ground (CG) flashes, after which intracloud (IC) flashes also occurred between middle and upper levels of the storm. The model simulation reproduces the basic evolution of lightning from low and middle levels to upper levels. The observed lightning indicated an initial charge structure of at least an inverted dipole (negative charge above positive). The simulations show that noninductive charge separation higher in the storm can enhance the main negative charge sufficiently to produce negative CG flashes before upper-level IC flashes commence. The result is a “bottom-heavy” tripole charge structure with midlevel negative charge and a lower positive charge region that is more significant than the upper posit...

/pdf/simulated-electrification-of-a-small-thunderstorm-with-two-4xi7tmq0p7.pdf

Simulated Electrification of a Small Thunderstorm with Two-Moment Bulk Microphysics

https://colonius.caltech.edu/pdfs/JohnsenColonius2006.pdf

Implementation of WENO schemes in compressible multicomponent flow problems

Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

In recent years high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today''s computers. In this paper we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user''s point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.

https://www.cs.odu.edu/~mln/ltrs-pdfs/icase-2001-11.pdf

High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD

In this paper, we discuss the local discontinuous Galerkin methodscoupled with two specific explicit-implicit-null time discretizations forsolving one-dimensional nonlinear diffusion problems $U_t=(a(U)U_x)_x$.The basic idea is to add and subtract two equal terms $a_0~U_{xx}$on the right-hand side of the partial differential equation,then to treat the term $a_0~U_{xx}$ implicitly and the other terms$(a(U)U_x)_x-a_0~U_{xx}$ explicitly. We give stability analysis for themethod on a simplified model by the aid of energy analysis,which gives a guidance for the choice of $a_0$, i.e., $a_0~\ge~\max\{a(u)\}/2$to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model, andnumerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.

Local discontinuous Galerkin methods with explicit-implicit-nulltime discretizations for solving nonlinear diffusion problems

In this talk we discuss the construction of efficient implementation of ENO (essentially non-oscillatory) schemes approximating systems of hyperbolic conservation laws of the type: 

$${u_t} + \sum\limits_{i = 1}^d {{f_i}} {\left( u \right)_{{x_i}}} = 0{\text{ (or = g(u,x,t) a forcing term)}}$$

(1a)



$$u\left( {x,\left. 0 \right)} \right. = {u^0}\left( x \right)$$

(1b)

where u = (u1 , ... , um) T , x = (xl, x2, ... , x d ), and, for real ξ = (ξ1,..., ξ d ), the combination $\sum\nolimits_{i = 1}^d \xi \frac{{\partial {f_i}}}{{\partial u}}$ always has m real eigenvalues and a complete set of eigenvectors. We use the notations xJ = J • Δx, tn = nΔt, and use u J n to denote the computed approximation to the exact solution u(xJ,t) of (1). We use bold face letters for vectors and plain letters scalars.

Shu Chi-Wang

Papers

High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD

Local discontinuous Galerkin methods with explicit-implicit-nulltime discretizations for solving nonlinear diffusion problems

Recent Progress On Essentially Non-Oscillatory Shock Capturing Schemes