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The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom, the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity (AAV) is proposed and combined with the displacement shallow water wave equation (DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effectiveness of the proposed AAV is first illustrated by using the steady-state solution and the small perturbation analysis. Then, the action mechanism of the AAV on the shallow water waves with the uneven bottom is explained by using the Fourier theory. It is shown that the AVV can suppress the wave with the large wave number, and can also suppress the numerical oscillations for the rapidly varying bottom. Finally, four numerical examples are given, and the numerical results show that the DSWWE combined with the AAV can effectively simulate the shock waves, accurately capture the movements of wet-dry interfaces, and precisely preserve the mass. [ABSTRACT FROM AUTHOR]

A finite-difference algorithm is proposed for numerical modeling of hydrodynamic flows with rarefaction shocks, in which the fluid undergoes a jump-like liquid-gas phase transition. This new type of flow discontinuity, unexplored so far in computational fluid dynamics, arises in the approximation of phase-flip (PF) hydrodynamics, where a highly dynamic fluid is allowed to reach the innermost limit of metastability at the spinodal, upon which an instantaneous relaxation to the full phase equilibrium (EQ) is assumed. A new element in the proposed method is artificial kinetics of the phase transition, represented by an artificial relaxation term in the energy equation for a "hidden" component of the internal energy, temporarily withdrawn from the fluid at the moment of the PF transition. When combined with an appropriate variant of artificial viscosity in the Lagrangian framework, the latter ensures convergence to exact discontinuous solutions, which is demonstrated with several test cases. [ABSTRACT FROM AUTHOR]

The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement. [ABSTRACT FROM AUTHOR]

Based on the second viscosity, the local differential quadrature (LDQ) method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to simulate shock tubes based on the viscosity model. The roles of the shear viscous stress and the second viscous stress are checked. The results show that the viscosity model combined with the LDQ method can capture the main characteristics of shocks, and this technique is objective and simple. [ABSTRACT FROM AUTHOR]