In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code

This paper provides a users’ guide to a new, general finite difference method for the numerical solution of systems of convection dominated conservation laws. We include both extensive motivation for the method design, as well as a detailed formulation suitable for direct implementation. Essentially Non-Oscillatory (ENO) methods are a class of high accuracy, shock capturing numerical methods for hyperbolic systems of conservation laws, based on upwind biased differencing in local characteristic fields. The earliest ENO methods used control volume discretizations, but subsequent work [12] has produced a simpler finite difference form of the ENO method. While this method has achieved excellent results in a great variety of compressible flow problems, there are still special situations where noticeable spurious oscillations develop. Why this occurs is not always understood, and there has been no elegant way to eliminate these problems. Based on the extensive work of Donat and Marquina [1], it appears that these difficulties arise from using a single transformation to local characteristic

We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235--276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.

In many problems of interest, solid objects are treated as rigid bodies in compressible flowfields. When these solid objects interact with certain features of the compressible flowfield, inaccurate solutions may develop. In particular, the well known "overheating effect" occurs when a shock reflects off of a stationary solid wall boundary causing overshoots in temperature and density, while pressure and velocity remain constant (see e.g. [3, 7, 13, 14]). This "overheating effect" is more dramatic when compressible flows are coupled to moving solid objects (e.g. moving pistons), where the nonphysical density and temperature overshoots can be cumulative and lead to negative values. We consider the general class of material interface problems where numerical methods can predict pressure and velocity adequately, but fail miserably in their prediction of density and temperature. Motivated by both total variation considerations and physical considerations, we have developed a simple but gene...

Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.3 Classification of differential equations : : : : : : : : : : : : : : : : : : : : : : : 7 2. Derivation of conservation laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 The Euler equations of gas dynamics : : : : : : : : : : : : : : : : : : : : : : : 13 2.2 Dissipative fluxes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Source terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.4 Radiative trans

Shock waves in granular gases generated by either a vertically vibrated granular layer or by hitting an obstacle at rest are treated by means of a shock capturing scheme that approximates the Euler equations of granular gas dynamics with an equation of state (EOS), introduced by Goldshtein and Shapiro [ J. Fluid Mech. 282 (1995) 75], that takes into account the inelastic collisions of granules. We include a sink term in the energy balance to account for dissipation of the granular motion by collisional inelasticity, proposed by Ha# [J. Fluid Mech. 134 (1983) 401], and the gravity field added as source terms. We have implemented an approximate Riemann solver, due to the second author [J. Comput. Phys. 125 (1996) 42], that works robust under low granular temperatures, high Mach numbers and near close-packed limit, damping post-shock oscillations. We have performed several numerical tests to show numerical evidence of the above features. We have computed the approximate solution to the following problems: a one-dimensional granular gas falling on a plate under the acceleration of gravity until close-packed limit, various one-dimensional blast waves evolving in time in the absence of gravity, a one-dimensional vertically vibrated granular layer under a sinusoidal perturbation and the two-dimensional reflecting shock wave generated when granular gas hits an angular obstacule through the acceleration of gravity.

An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327-353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and high-resolution shockcapturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary one-dimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package clawpack and is freely available on the web.

In this paper we are going to study the gas evolution dynamics of the exact and approximate Riemann solvers, e.g., the Flux Vector Splitting (FVS) and the Flux Difference Splitting (FDS) schemes. Since the FVS scheme and the Kinetic Flux Vector Splitting (KFVS) scheme have the same physical mechanism and similar flux function, based on the analysis of the discretized KFVS scheme the weakness and advantage of the FVS scheme are clearly observed. The subtle dissipative mechanism of the Godunov method in the 2D case is also analyzed, and the physical reason for shock instability, i.e., carbuncle phenomena and odd-even decoupling, is presented.

Flux-gradient and source term balancing for certain high resolution shock-capturing schemes

As a continuous effort to understand the Godunov-type schemes, following the paper "Projection Dynamics in Godunov-Type Schemes" [29] (Jcp Vol.142, 412-427, 1998), in this paper we are going to study the gas evolution dynamics of the exact and approximate Riemann solvers. More specifically, the underlying dynamics of Flux Vector Splitting (FVS) and Flux Difference Splitting (FDS) schemes will be analyzed. Since the FVS scheme and the Kinetic Flux Vector Splitting (KFVS) scheme have the same physical mechanism and numerical formulations, based on the governing equation of the discretized KFVS scheme, the weakness and advantages of FVS scheme are clearly observed. Also, in this paper, the implicit equilibrium assumption in the Godunov flux will be analyzed. Due to the numerical shock thickness related to the cell size, the numerical scheme should be able to capture both equilibrium and non-equilibrium flow behavior in smooth and discontinuous regions. The Godunov flux basically lacks the...