An adaptive ghost uidnite volume method is developed for one{ and two dimensional compressible multi-medium ows in this work. It couples the real ghostuid method (GFM) [SIAM J. Sci. Comput. 28 (2006) 278] and the adaptive moving mesh method [SIAM]. Numer. Anal. 41 (2003) 487; J. Comput. Phys. 188 (2003) 543], and thus retains their advantages. This work shows that the local mesh clus-tering in the vicinity of the material interface can efiectively reduce both numerical and conservative errors caused by the GFM around the material interface and other discontinuities. Besides the improvement of oweld resolution, the adaptive ghost uid method also largely increases the computational eciency. Several numerical experiments are conducted to demonstrate robustness and e±ciency of the current method. They include several 1D and 2D gas-water problems, involving a large density gradient at the material interface and strong shock-interface interactions. The results show that our algorithm can capture the shock waves and the material interface accurately, and is stable and robust even solution with large density and pressure gradients

The original ghost fluid method (GFM) developed in [13] and the modifiedGFM (MGFM) in [26] have provided a simple and yet flexible way to treat twomediumflow problems. The original GFM and MGFM make the material interface "invisible" during computations and the calculations are carried out as for a singlemedium such that its extension to multi-dimensions becomes fairly straightforward.The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservationlaws is a high order accurate finite element method employing the usefulfeatures from high resolution finite volume schemes, such as the exact or approximateRiemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper,we investigate using RKDG finite element methods for two-medium flow simulationsin one and two dimensions in which the moving material interfaces is treated via nonconservativemethods based on the original GFM and MGFM. Numerical results forboth gas-gas and gas-water flows are provided to show the characteristic behaviors ofthese combinations.

The dynamic response of deformable structures subjected to shock load and cavitation reload has been simulated using a multiphase model, which consists of an interface capturing method and a one-fluid cavitation model. Fluid-structure interaction (FSI) is captured via a modified ghost fluid method (Liu et al. in J Comput Phys 190: 651-681, 2003), where the structure is assumed to be a hydro-elasto-plastic material if subjected to a strong shock load. Bulk cavitation near the structural surface is captured using an isentropic model (Liu et al. in J Comput Phys 201:80–108, 2004). The integrated multiphase model is validated by comparing numerical predictions with 1D analytical solutions, and with numerical solutions calculated using the cavitation acoustic finite element (CAF?) method (Sprague and Geers in Shocks vib 7: 105-122, 2001). To assess the ability of the multiphase model for multi-dimensions, underwater explosions (UNDEX) near structures are computed. The importance of cavitation reloading and FSI is investigated. Comparisons of the predicted pressure time histories with different explosion center are shown, and the effect on the structure is discussed.

Previous numerical tests have shown that the modified ghost fluid method (MGFM) [T.G. Liu, B.C. Khoo, K.S. Yeo, Ghost fluidmethod for strong shock impacting on material interface, J. Comput. Phys. 190 (2003) 651–681; T.G. Liu, B.C. Khoo, C.W.Wang,The ghost fluid method for compressible gas-water simulation, J. Comput. Phys. 204 (2005) 193-221] is robust and performs muchbetter than the original GFM [R.P. Fedkiw, T. Aslam, B. Merriman, S. Osher, A non-oscillatory Eulerian approach to interfaces inmultimaterial flows (the ghost fluid method), J. Comput. Phys. 152 (1999) 457-492]. In this work, a rigorous analysis is carriedout on the accuracy of the MGFM when applied to the gas-gas Riemann problem. It is shown that at the material interface theMGFM solution approximates the exact solution to at least second-order accuracy in the sense of comparing to the exact solutionof a Riemann problem. On the other hand, the results by the original GFM have generally no-order accuracy if the interface is notin normal motion.

The Runge–Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high orderfinite element method, which utilizes the useful features from high resolution finite volume schemes, such as the exactor approximate Riemann solvers, TVD Runge–Kutta time discretizations, and limiters. In this paper, we investigate usingthe RKDG finite element method for compressible two-medium flow simulation with conservative treatment of the movingmaterial interfaces. Numerical results for both gas–gas and gas–water flows in one-dimension are provided to demonstratethe characteristic behavior of this approach.

This note gives a numerical investigation for the popular high resolution conservativeschemes when applied to inviscid, compressible, perfect gas °ows with an initial highdensity ratio as well as a high pressure ratio. The results show that they work veryine±ciently and may give inaccurate numerical results even over a very ?ne meshwhen applied to such a problem. Numerical tests show that increasing the order ofaccuracy of the numerical schemes does not help much in improving the numericalresults. How to cure this di±culty is still open.

An underwater explosion near a free surface constitutes an explosive gas–water–air system with a shock and free surface interactionand the presence of bulk cavitation region. This paper applies a simplified modified ghost fluid method [T.G. Liu, et al., Comm.Comput. Phys. 1 (2006) 898] to simulate the explosive gas–water and water–air interfaces and an isentropic one-fluid cavitationmodel [T.G. Liu, et al., J. Comput. Phys. 201 (2004) 80] to describe and capture the unsteady cavitation just below the free surface.It is found that the proposed ghost fluid method can accurately simulate the gas–water/water–air compressible flows with the waveinteraction at the interfaces and the deformation of the free surface. The isentropic one-fluid cavitation model, on the other hand,is capable of simulating the dynamic creation and evolution of the bulk cavitation below the free surface.

We propose a methodology based on the boundary element method (BEM) to simulate pressure pulse–bubble interaction. The pulseresembles a shock wave and is in the form of a step pulse function incorporated into the Bernoulli equation. Compressibility effects of thewater surrounding the bubble are neglected, and the dynamic response of the bubble to the impinging pulse is assumed to be mainlyinertia controlled. The interaction induces the formation of a high-speed jet that penetrates the bubble. Results show that bubble shape,collapse time and jet velocity are in good agreement with other numerical models and experiments, and the method is more computationallyefficient.

An analysis is carried out for the ghost fluid method (GFM) based algorithm as applied to the gas–waterRiemann problems, which can be construed as two single-medium GFM Riemann problems. It is found that theinability to provide correct and consistent Riemann waves in the respective real fluids by these two GFM Riemannproblems may lead to inaccurate numerical results. Based on this finding, two conditions are suggested and imposedfor the ghost fluid status in order to ensure that correct and consistent Riemann waves are provided in the respectivereal fluids during the numerical decomposition of the singularity. Using these two conditions to analyse some of theexisting GFM-based algorithms such as the original GFM [J. Comput. Phys. 152 (1999) 457], the new version GFM[J. Comput. Phys. 166 (2001) 1; J. Comput. Phys. 175 (2002) 200] and the modified GFM (MGFM) [J. Comput.Phys. 190 (2003) 651], it is found that there are ranges of conditions for each type of solution where either the originalGFM or the new version GFM or both are unable to provide correct or consistent Riemann waves in one ofthe real fluids. Within these ranges, examples can be found such that either the original GFM or the new versionGFM or both are unable to provide accurate results. The MGFM is also found to encounter difficulties whenapplied to nearly cavitating flow. Various examples are presented to demonstrate the conclusions obtained. TheMGFM with proposed modification when applied to nearly cavitating flow is then found to be quite robust andcan provide relatively reasonable results.