High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

2009/02/05 by Chi-Wang Shu, Chi‐Wang Shu · 957 citations
Earth and Planetary Sciences · Engineering · Mathematics · #Applied mathematics #Classification of discontinuities #Computational Fluid Dynamics and Aerodynamics #Computer science #Discontinuity (linguistics) #Flow (mathematics) #Fluid Dynamics and Turbulent Flows #Geometry #Hyperbolic partial differential equation #Mathematical analysis #Mathematical optimization #Mathematics #Meteorological Phenomena and Simulations #Partial differential equation

paper · doi:10.1137/070679065

published in SIAM Review 51(1), 82-126 (Society for Industrial and Applied Mathematics)

openalex publication_date 2009/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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.

Citations

• On the Gibbs Phenomenon and Its Resolution
• A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow
• The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V
• Strong Stability-Preserving High-Order Time Discretization Methods
• Multiresolution algorithms for the numerical solution of hyperbolic conservation laws
• Efficient implementation of essentially non-oscillatory shock-capturing schemes

Cited by

• Hybridizing WENO implementations of interpolation and reconstruction-wise operation for upwind-biased schemes with free-stream preservation
• Implicit-explicit schemes for compressible Cahn-Hilliard-Navier-Stokes equations on staggered grids
• Kinetic-controlled hydrodynamics for multilane traffic models
• Fast sparse grid simulations of fifth order WENO scheme for high dimensional hyperbolic PDEs
• A Sweeping Positivity-Preserving High Order Finite Difference WENO Scheme for Euler Equations
• How does ice shell geometry shape ocean dynamics on icy moons?
• Towards Exascale Simulations of the ICM Dynamo with WENO-Wombat
• A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation
• A DSMC-PIC coupling method for the Vlasov-Maxwell-Landau system
• On free-stream preservation in stationary grids for arbitrary linear upwind schemes
• An Adaptive Finite Element Method for Marker-Driven Level-Set Transport on Hierarchical Meshes
• Back reaction of the untwisting solar corona scars sunspots
• Exact Total Variation Minimizers as Non-Oscillatory Limiters in High-Order Methods for Conservation Laws
• A Nodal Discontinuous Galerkin Method with Low-Rank Velocity Space Representation for the Multi-Scale BGK Model
• Compact Higher-order Gas-kinetic Schemes with Spectral-like Resolution for Compressible Flow Simulations
• A Kernel Based High Order "Explicit" Unconditionally Stable Constrained Transport Method for Ideal Magnetohydrodynamics
• A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
• Robust Spectral Solver for High-Fidelity Investigations of Aerospike Nozzle Flow Dynamics
• Task-Based Programming for Adaptive Mesh Refinement in Compressible Flow Simulations
• The coherent structures of EVP Fluid Flow Past a Circular Cylinder
• Central Runge-Kutta discontinuous Galerkin methods for the special relativistic hydrodynamics
• Conservative Multi-Dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations
• High order maximum principle preserving discontinuous Galerkin method for convection-diffusion equations
• A generalized ENO reconstruction in compact GKS for compressible flow simulations
• Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations
• Sharpening methods for finite volume schemes
• Linear and nonlinear waves
• A General Framework to Derive Linear, Decoupled and Energy-stable Schemes for Reversible-Irreversible Thermodynamically Consistent Models: Part I Incompressible Hydrodynamic Models
• One- and multi-dimensional CWENOZ reconstructions for implementing boundary conditions without ghost cells
• Kernel Based High Order "Explicit" Unconditionally-Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations
• Implicit-Explicit WENO schemes for the equilibrium dispersive model of chromatography
• Fourth- and Higher-Order Semi-Lagrangian Finite Volume Methods for the Two-dimensional Advection Equation on Arbitrarily Complex Domains
• A kernel-based discretisation method for first order partial\n differential equations of evolution type
• High-order well-balanced finite volume schemes for the Euler equations\n with gravitation
• A high-order finite difference WENO scheme for ideal magnetohydrodynamics on curvilinear meshes
• Direct Numerical Simulations of pore competition in idealized micro-spall using the VOF method
• A New Class of High-Order Methods for Fluid Dynamics Simulations using Gaussian Process Modeling
• A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations
• Discontinuous Galerkin Semidiscretization of the Information Geometric Regularized Compressible Euler Equations
• A high-order Newton multigrid method with a simplified Jacobian for steady-state shallow water equations
• Analysis of a generalised Boltzmann equation for anomalous diffusion under time-dependent fields
• Addressing the issue of mass conservation error and the connected problem of Carbuncle formation
• WENO interpolations and reconstructions using data bounded polynomial\n approximation
• A Weighted Upwind Vector Kinetic Lattice Boltzmann Method For Hyperbolic Conservation Laws
• Numerical solution of the Burgers' equation with high order splitting\n methods
• MORPHOLOGY AND DYNAMICS OF SOLAR PROMINENCES FROM 3D MHD SIMULATIONS
• GPU-based compressible lattice Boltzmann simulations on non-uniform grids using standard C++ parallelism: From best practices to aerodynamics, aeroacoustics and supersonic flow simulations
• Initiation Route of Coronal Mass Ejections: II. The Role of Filament Mass
• Adaptive interpolation with maximum order close to discontinuities
• Comparison of improved finite-difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows
• Free-stream preserving finite difference schemes for ideal magnetohydrodynamics on curvilinear meshes
• A Multistate Low-dissipation Advection Upstream Splitting Method for Ideal Magnetohydrodynamics
• Model-based assessment of the impact of driver-assist vehicles using kinetic theory
• Model-Based Design of Long-Distance Tracer Transport Experiments in Plants. [europepmc]
• Higher-order accurate space-time schemes for computational astrophysics-Part I: finite volume methods. [europepmc]
• Hierarchical modeling of mechano-chemical dynamics of epithelial sheets across cells and tissue. [europepmc]
• Well-balanced methods for computational astrophysics. [europepmc]
• General hydrodynamic features of elastoviscoplastic fluid flows through randomised porous media. [europepmc]
• The coherent structures of EVP fluid flow past a circular cylinder. [europepmc]

Related

• A New Adaptive Weighted Essentially Non-Oscillatory WENO-θ Scheme for Hyperbolic Conservation Laws
• Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator
• Third order WENO scheme on sparse grids for hyperbolic equations
• A spatial-temporal weight analysis and novel nonlinear weights of weighted essentially non-oscillatory schemes for hyperbolic conservation laws
• Fast sparse grid simulations of fifth order WENO scheme for high dimensional hyperbolic PDEs