Hyperbolic equation shocks

There is not enough space to describe the properties of this equation. Suffice it to say that the Buckley-Leverett equation has shock-like solutions, where the saturation front is a wave propagating through the reservoir. This combination of an elliptic equation for the total pressure and a parabohc, but nearly hyperbolic equation for the saturation, gives rise to great mathematical interest in two-phase flow though porous media. 

Due to the hyperbolicity and nonlinearity of the model equations, associated with possible shocks in granular flows over non-trivial topography, numerical solutions with the traditional high-order accuracy methods are often accompanied with numerical oscillations of the depth profile and velocity field. This usually leads to numerical instabilities unless these are properly counteracted by a sufficient amount of artificial numerical diffusion. Here, a non-oscillatory central (NOC) difference scheme with a total variation diminishing (TVD) limiter for the cell reconstruction is employed, see e.g. [4], [12] we obtain numerical solutions without spurious oscillations. In order to test the model equations, we consider an ideal mountain subregion in which the talweg is defined by the slope function... 

In 1959, Godunov [64] introduced a novel finite volume approach to compute approximate solutions to the Euler equations of gas d3mamics that applies quite generally to compute shock wave solutions to non-linear systems of hyperbolic conservation laws. In the method of Godunov, the numerical approximation is viewed as a piecewise constant function, with a constant value on each finite volume grid cell at each time step and the time evolution... 

When the NDF represents the particle velocity, the transport equation is hyperbolic and thus the solution need not be smooth and need not be differentiable at every point in space. In fact, the treatment of shocks in the NDF is a significant challenge. 

In summary, although the weakly hyperbolic nature of Eq. (8.7) has been shown rigorously only for ID phase space (i.e. one velocity component in the KE), experience strongly suggests that the full 3D system is also weakly hyperbolic. This observation implies that the numerical schemes used to solve the moment-transport equations closed with QBMM must be able to handle local delta shocks in the moments. Qne such class of numerical schemes consists of the kinetics-based finite-volume solvers presented in Section 8.2. As a final note, we should mention that the work of Chalons et al (2012) using extended Gaussian quadrature (see Section 3.3.2) and kinetics-based finite-volume solvers to close Eq. (8.7) suggests that the system with 2A + 1 moments is fully hyperbolic and thus does not exhibit... 

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