We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order $O(\delta _1^{4N+2}+\delta _2^{4N+2})$ in the sense of consistency, where $\delta _1$ and $\delta _2$ are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and $N$ is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.

The modified ghost fluid method (MGFM), due to its reasonable treatment for ghost fluid state, has been shown to be robust and efficient when applied to compressible multi-medium flows. Other feasible definitions of the ghost fluid state, however, have yet to be systematically presented. By analyzing all possible wave structures and relations for a multi-medium Riemann problem, we derive all the conditions to define the ghost fluid state. Under these conditions, the solution in the real fluid region can be obtained exactly, regardless of the wave pattern in the ghost fluid region. According to the analysis herein, a practical ghost fluid method (PGFM) is proposed to simulate compressible multi-medium flows. In contrast with the MGFM where three degrees of freedomat the interface are required to define the ghost fluid state, only one degree of freedomis required in this treatment. However, when these methods proved correct in theory are used in computations for the multi-medium Riemann problem, numerical errors at the material interface may be inevitable. We show that these errors are mainly induced by the single-medium numerical scheme in essence, rather than the ghost fluid method itself. Equipped with some density-correction techniques, the PGFM is found to be able to suppress these unphysical solutions dramatically.

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

We present a new numerical method to approximate the solutions of an Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity source term involved in the latter equations. In order to approximate this source term, its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The method has been implemented in the software HERACLES [29] and several numerical experiments involving gravitational flows for astrophysics highlight the scheme.

The space-time conservation element and solution element (CE/SE) method is proposed for solving a conservative interface-capturing reduced model of compressible two-fluid flows. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term for accounting the energy exchange. The one and two-dimensional flow models are numerically investigated in this manuscript. The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building block of the suggested method. The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation. In order to reveal the efficiency and performance of the approach, several numerical test cases are presented. For validation, the results of the current method are compared with other finite volume schemes.

This paper is an introduction to a conservative, positive numerical scheme which takes into account the phenomena of reflection and transmission of high frequency acoustic waves at a straight interface between two homogeneous media. Explicit forms of the interpolation coefficients for reflected and transmitted wave vectors on a two-dimensional uniform grid are derived. The propagation model is a Liouville transport equation solved in phase space.

We use the compensated compactness method coupled with some basic ideas of kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global bounded entropy solutions to the Le Roux system. This new method of the reduction of Young measures can be applied to solve other problems.