Shock Capturing Methods - Classical and Modern Shock Capturing Methods

Classical and Modern Shock Capturing Methods

From an historical point of view, shock-capturing methods can be classified into two general categories: viz., classical methods and modern shock capturing methods (also called high-resolution schemes). Modern shock-capturing methods are generally upwind based in contrast to classical symmetric or central discretization. Upwind-type 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 shock-capturing 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 shock-capturing methods, numerical dissipation terms are usually linear and the same amount is uniformly applied at all grid points. Classical shock-capturing methods only exhibit accurate results in the case of smooth and weak-shock solution, but when strong shock waves are present in the solution, non-linear instabilities and oscillations can arise across discontinuities. Modern shock-capturing methods have, however, a non-linear 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 shock-capturing methods include the MacCormack method (uses a discretization scheme for the numerical solution of hyperbolic partial differential equations), Lax–Wendroff method (based on finite differences, uses a numerical method for the solution of hyperbolic partial differential equations), and Beam–Warming method. Examples of modern shock-capturing schemes include, higher-order total variation diminishing (TVD) schemes first proposed by Harten, flux-corrected transport scheme introduced by Boris and Book, Monotonic Upstream-centered Schemes for Conservation Laws (MUSCL) based on Godunov approach and introduced by van Leer, various Essentially Non-Oscillatory 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 non-linear switch functions, fall in between the classical and modern shock-capturing methods.