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Shock capturing methodsIn computational fluid dynamics, shockcapturing methods are a class of techniques for computing inviscid flows with shock waves. Computation of flow through shock waves is an extremely difficult task because such flows results in sharp, discontinuous changes in flow variables pressure, temperature, density, and velocity across the shock. Additional recommended knowledge
ExplanationIn shockcapturing approach the governing equations of inviscid flows (Euler equations) are cast in conservation form and any shock waves or discontinuities are computed as part of the solution. Here, no special treatment is employed to take care of the shocks themselves. This is in contrast to the shockfitting method, where shock waves are explicitly introduced in the solution using appropriate shock relations (RankineHugoniot relations). The shock capturing methods are relatively simple compared to the more elaborate shock fitting methods. However, the shock waves predicted by shockcapturing methods are generally not sharp and smear over several grid points. Also, classical shockcapturing methods have the disadvantages that unphysical oscillations (Gibbs phenomenon) may develop in the vicinity of strong shocks. Euler equationThe Euler equations are the governing equations for inviscid flows. To implement shockcapturing methods, the conservation form of the Euler equations are used. For a flow without external heat transfer and work transfer (isoenergetic flow), the conservation form of the Euler equation in Cartesian coordinate system can be written as where the vectors U, F, G, and H are given by where e_{t} is the total energy (internal energy + kinetic energy + potential energy) per unit mass. That is The Euler equation may be integrated with any of the shockcapturing methods available to obtain the solution. Classical and modern shock capturing methodsFrom an historical point of view, shockcapturing methods can be classified into two general categories: viz., classical methods and modern shock capturing methods (also called highresolution schemes). Modern shockcapturing methods are generally upwind based in contrast to classical symmetric or central discretization. Upwindtype differencing schemes attempt to discretize hyperbolic partial differential equations by using differencing biased in the direction determined by the sign of the characteristic speeds. On the other hand, symmetric or central schemes do not consider any information about the wave propagation in the discretization. No matter what type of shockcapturing scheme is used, a stable calculation in presence of shock waves requires a certain amount of numerical dissipation, in order to avoid the formation of unphysical numerical oscillations. In the case of classical shockcapturing methods, numerical dissipation terms are usually linear and the same amount is uniformly applied at all grid points. Classical shockcapturing methods only exhibit accurate results in the case of smooth and weakshock solution, but when strong shock waves are present in the solution, nonlinear instabilities and oscillations can arise across discontinuities. Modern shockcapturing methods have, however, a nonlinear numerical dissipation, with an automatic feedback mechanism which adjusts the amount of dissipation in any cell of the mesh, in accord to the gradients in the solution. These schemes have proven to be stable and accurate even for problems containing strong shock waves. Some of the well known classical shockcapturing methods include the MacCormack method, LaxWendroff method, and BeamWarming method. Examples of modern shockcapturing schemes include, higher order Total Variation Diminishing (TVD) schemes first proposed by Harten, FluxCorrected Transport scheme introduced by Boris and Book, Monotonic Upstreamcentered Schemes for Conservation Laws (MUSCL) based on Godunov approach and introduced by van Leer, various Essentially NonOscillatory schemes (ENO) proposed by Harten et al., and Piecewise Parabolic Method (PPM) proposed by Woodward and Colella. Another important class of high resolution schemes belongs to the approximate Riemann solvers proposed by Roe and by Osher. The schemes proposed by Jameson and Baker, where linear numerical dissipation terms depend on nonlinear switch functions, fall in between the classical and modern shockcapturing methods. ReferencesBooks
Technical Papers


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Shock_capturing_methods". A list of authors is available in Wikipedia. 