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In this work, we show that the modified ghost fluid method might suffer overheating and leads to inaccurate numerical results when directly applied to a moving rigid boundary with acceleration. We discover the insightful reasons and then develop a new technique to take into account the effect of boundary acceleration on the definition of ghost fluid states based on a generalized Piston–Riemann problem. Theoretical analysis and numerical results show that the modified ghost fluid method with acceleration correction can overcome such difficulty effectively.

In this paper, a third-order reconstructed DG( p 1 p 2 ) method based on the variational reconstruction (VR) (Wang et al., 2017 [24] ) is developed for simulating the two dimensional steady-state compressible flows on unstructured and hybrid grids. The proposed method combines the advantages of the DG discretization with the flexibility of the variational reconstruction, which exhibits its superior potential in enhancing the level of accuracy compared to the underlying DG method. In this variational rDG( p 1 p 2 ) method, the low order degrees of freedom are evolved through the underlying DG( p 1 ) method, while the high order degrees of freedom are reconstructed through the variational reconstruction, in which the constitutive relations are built by minimizing the so-called ‘cost function’. The cost function is defined by the total interfacial jump integration in the computational domain using the variational method. The large sparse linear system resulted by the variational reconstruction is solved in an efficient way coupled with the temporal discretization for the steady-state simulations. A number of test cases are presented to assess the performance of the new high order variational rDG( p 1 p 2 ) method.

In this work, an adjoint-based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous Galerkin method for compressible Euler equations to investigate the significance of each design variable of airfoil B-spline parameterization. We first parameterize the airfoil by B-spline curve approximation with some control points viewed as design variables, and build the B-spline representation of the flow field with the curve to apply the goal-oriented h -adaptive isogeometric DG method for flow solution. Then we compute and employ the discrete adjoint solutions for both multi-target error estimation in adaptive mesh refinement. With the isogeometric nature, not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables, indicating that the numerical solutions and objective could be differentiable with respect to those variables. Consequently, the gradient is totally computed in an accurate approach, and the sensitivity analysis is thus improved, by reducing the spatial discretization error and introducing the analytical expression of derivative, to reveal the key parameters for optimization in an intuitive and efficient manner. Although the SQP optimization algorithm is adopted in the paper, the given accurate gradient can be applied to any gradient-based optimization algorithm. The proposed algorithm is demonstrated on RAE2822 airfoil with inviscid transonic flow, where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area. The numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal.

In this work, a solution remapping technique is developed to accelerate the flow convergence for the intermediate shapes when a high-order discontinuous Galerkin (DG) method is employed as a compressible Euler flow solver in the airfoil design problems. Once the shape is updated, the proposed technique is applied to initialize the flow simulation for the new shape via a solution remapping formula and a maximum-and-minimum-preserving limiter. First, the solution remapping formula is used to remap the solution of the current shape into a piecewise polynomial on the mesh of the new shape. Then the piecewise polynomial is constrained with the maximum-and-minimum-preserving limiter. The modified piecewise polynomial is used as the initial value for the new shape. Numerical experiments show that the proposed technique can attractively accelerate flow convergence and significantly reduce up to 80% of the computational time in the airfoil design problems with a high-order DG solver.

Level set (LS) method is a widely used interface capturing method. In the simulations of incompressible two-phase flows, in order to avoid discontinuities at interfaces, the LS function is usually taken as a smeared-out Heaviside function bounded on [0, 1] and advected by a given velocity field $$\mathbf {u}$$ obtained from the solution of the incompressible Navier-Stokes equations. In the incompressible limit $$\nabla \cdot \mathbf {u}=0$$ , the advection equation for the LS function can be written and discretized in conservative form. However, due to numerical errors, the resulting velocity field is in general not divergence free which leads to the solution of the advection equation in conservative form does not satisfy the maximum principle. To overcome this issue, in this work, we develop a high-order discontinuous Galerkin (DG) method to directly solve the advection equation for the LS function in non-conservative form. Moreover, we prove that by applying a linear scaling limiter, the proposed method together with a strong stability preserving (SSP) time discretization scheme can satisfy the strict maximum principle under a suitable CFL condition. Numerical simulations of several well-known benchmark problems, including the application to incompressible two-phase flows, are presented to demonstrate the high-order accuracy and maximum-principle-satisfying property of the proposed method.

In this work, we develop a quasi-conservative discontinuous Galerkin method for the simulation of compressible gas-gas and gas-water two-medium flows by solving the five-equation transport model. This spatial discretization is a direct extension of the quasi-conservative finite volume discretization to the discontinuous Galerkin framework, thus, preserves uniform velocity and pressure fields at an isolated material interface. Furthermore, for discontinuities with a large pressure ratio, low density, and a dramatic change of material property where nonphysical values may occur, a strategy for imposing the bound-preserving limiting for volume fraction and a positivity-preserving limiting for density of each fluid and internal energy is developed and analyzed based on the quasi-conservative DG( $$p_1$$ ) discretization. Typical test cases for both one- and two-dimensional problems are provided to demonstrate the performance of the proposed method.

In this work, we use exact solutions of one-dimensional Burgers equation to train an artificial neuron as a shock wave detector. The expression of the artificial neuron detector is then modified into a practical form to reflect admissible jump of eigenvalues. We show the working mechanism of the practical form is consistent with compressing or intersecting of characteristic curves. In addition, we prove there is indeed a discontinuity inside the cell detected by the practical form, and smooth extrema and large gradient regions are never marked. As a result, we apply the practical form to numerical schemes as a shock wave indicator with its easy extension to multi-dimensional conservation laws. Numerical results are present to demonstrate the robustness of the present indicator under Runge–Kutta Discontinuous Galerkin framework, its performance is generally compared to TVB-based indicators more efficiently and accurately. To treat the initial inadmissible jumps, including linear contact discontinuities and those evolving into rarefaction waves, a preliminary strategy of combining a traditional indicator in the beginning with the present indicator is suggested. We believe the present indicator can be applied to unstructured mesh in the future.

In general, it needs to take about nearly 10 grid points inside a wall boundary layer for low accuracy order methods to get satisfactory wall-normal gradient related results, such as friction coefficient and wall heat flux. If there exist extreme points inside the boundary layer, this situation becomes even worse. In this work, with the help of the analytic solution of the one-dimensional steady compressible Navier-Stokes equations under some restrictions, we show that the flow variables actually vary in the form of an exponential function instead of a polynomial one inside the boundary layer. Then, we propose an exponential space for approximating the solutions inside the boundary layer and numerically implementing it in the frame of direct discontinuous Galerkin (DDG) method. We show that the DDG methods based on the exponential boundary-layer space give much better numerical results for both conservative variables and wall-normal gradients than those with the standard polynomial space. Generally only 1–2 grid points inside the boundary layer are demanded to resolve the wall boundary layer to obtain satisfactory wall-normal gradients under the exponential space. Preliminary extension to two-dimensional laminar boundary-layer flow shows a similar performance of the proposed exponential boundary-layer space, exhibiting its potential applications in high dimensions.

Usually, the temperature dependence of specific heats is neglected or the specific heats are frozen in interface computations for compressible two-material flows. In this paper, we present a practical interface treatment to faithfully capture the effect of high temperature on interface evolutions. A general technique for solving the Riemann problem equipped with a wide variety of equations of state (EOS) is established. In a unified framework for computing the interfacial states, it provides a convenient way to deal with the thermally perfect gas (PG) that considers the effect of temperature on specific heats. The algorithm of the complete and exact solution to Riemann problem with thermally PG is also designed in detail. Based on this technique, the modified ghost fluid method with an approximate Riemann solver is further extended to handle the interface of two-material flows involving thermally PG with chemical reactions. Several typical problems are selected to validate and test the present algorithm for the interaction between the thermally PG and other gases or liquids. The results indicate that the present algorithm enables an effective implementation for simulating various two-material multi-species flows with different types of EOS. As temperature increases, the behavior of the interfacial flows under the assumption of thermally PG EOS gradually differs from that under the assumption of calorically PG EOS.

In this work, an integral exact solution is proposed for one-dimensional elastic-perfectly plastic solid Riemann problem. Owing to the possible existence of elastic to plastic “phase” transition within the solid, the exact solution of its Riemann problem is much more complicated than that for the medium equipped with a uniform equation of state (EOS) or constitutive model. By constructing a five-equation hyperbolic governing system fully describing the nonlinear behavior of the solid in the Eulerian reference frame and scrutinizing every possible wave pattern of its Riemann problem, we acquire a complete list of exact solutions that contains as many as sixty-four different solution types neglecting the generation of vacuum. Each type of exact solutions is presented, and numerical simulations agree well with the obtained theoretical results.

In this paper, a class of new high order reconstructed DG (rDG) methods based on the compact least-squares (CLS) reconstruction [23] , [24] is developed for simulating the two dimensional steady-state compressible flows on hybrid grids. The proposed method combines the advantages of the DG discretization with the flexibility of the compact least-squares reconstruction, which exhibits its superior potential in enhancing the level of accuracy and reducing the computational cost compared to the underlying DG methods with respect to the same number of degrees of freedom. To be specific, a third-order compact least-squares rDG( p 1 p 2 ) method and a fourth-order compact least-squares rDG( p 2 p 3 ) method are developed and investigated in this work. In this compact least-squares rDG method, the low order degrees of freedom are evolved through the underlying DG( p 1 ) method and DG( p 2 ) method, respectively, while the high order degrees of freedom are reconstructed through the compact least-squares reconstruction, in which the constitutive relations are built by requiring the reconstructed polynomial and its spatial derivatives on the target cell to conserve the cell averages and the corresponding spatial derivatives on the face-neighboring cells. The large sparse linear system resulted by the compact least-squares reconstruction can be solved relatively efficient when it is coupled with the temporal discretization in the steady-state simulations. A number of test cases are presented to assess the performance of the high order compact least-squares rDG methods, which demonstrates their potential to be an alternative approach for the high order numerical simulations of steady-state compressible flows.

In this paper, an adjoint-based high-order h-adaptive direct discontinuous Galerkin method is developed and analyzed for the two dimensional steady state compressible Navier–Stokes equations. Particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous Galerkin discretizations: including the original direct discontinuous Galerkin method (DDG), the direct discontinuous Galerkin method with interface correction (DDG(IC)) and the symmetric direct discontinuous Galerkin method (SDDG). Theoretical analysis shows the extra interface correction term adopted in the DDG(IC) method and the SDDG method plays a key role in preserving the adjoint consistency. To be specific, for the model problem considered in this work, we prove that the original DDG method is not adjoint consistent, while the DDG(IC) method and the SDDG method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals. The performance of those three DDG methods is carefully investigated and evaluated through typical test cases. Based on the theoretical analysis, an adjoint-based h-adaptive DDG(IC) method is further developed and evaluated, numerical experiment shows its potential in the applications of adjoint-based adaptation for simulating compressible flows.

In a recent work Feng, Liu and Wang (2020) [10] , we imbedded characteristic compressing into an artificial neuron (AN) to propose a shock wave indicator on uniform mesh. In this work, the indicator is developed to unstructured grid. To achieve that, we retrain an AN on 1D randomly perturbed mesh, two prior information, (a) eigenvalue variable and (b) side-weighted average, is used in data pre-processing for reducing the influence of mesh size and keeping AN structure simple. The output of AN is then modified into a generalized and explicable form, which is used as the present shock wave indicator. We show that the troubled-cells detected by the present indicator include discontinuities caused by compressing of characteristic curves. The present indicator is then extended to multi-dimensional unstructured grid through constructing side-weighted average of eigenvalue on each spatial dimension. Numerical results are presented to demonstrate the performance of the present indicator combined with slope limiter and artificial viscosity, respectively, on various unstructured grids, the results show that the present indicator can detect shock and contact waves with low noise, and improves the indicating efficiency as well, the present indicator provides an attractive alternative in detecting shock waves on arbitrary grids and can be combined with various discontinuity-processing techniques.

In this paper, a residual-based h-adaptive high-order reconstructed DG (rDG) method based on the hybrid reconstruction strategy is developed for solving the compressible Euler equations on unstructured grids. The proposed method combines the advantages of high-order rDG discretization with appropriate residual-based error estimation techniques and h-adaptive refinement strategies, which exhibits its superior potential compared to the underlying DG methods. To be specific, a third-order hybrid rDG(p1p2) method has been carefully designed and evaluated on incompatible quadrilateral grids with hanging nodes in order to preserve 2-exactness property during the implementation of mesh refinement and coarsening. A residual-based error estimator is used as local error indicator during the h-adaptive procedures. A number of test cases are presented to assess the performance of the high-order h-adaptive rDG method. The hybrid reconstruction strategy combined with h-adaptive techniques presented in this work demonstrates promise for improving the level of accuracy and reducing the computational cost for numerical simulations of compressible inviscid flows.

A parallel, high-order direct Discontinuous Galerkin (DDG) method has been developed for solving the three dimensional compressible Navier-Stokes equations on 3D hybrid grids. The most distinguishing and attractive feature of DDG method lies in its simplicity in formulation and efficiency in computational cost. The formulation of the DDG discretization for 3D Navier-Stokes equations is detailed studied and the definition of characteristic length is also carefully examined and evaluated based on 3D hybrid grids. Accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions. The capability in handling curved boundary geometry is also demonstrated. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results obtained indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable results as the widely used BR2 scheme, clearly demonstrating that the DDG method provides an attractive alternative for solving the 3D compressible Navier-Stokes equations.

The equation of state (EOS) plays a crucial role in hyperbolic conservation laws for the compressible fluid. Whereas, the solid constitutive model with elastic-plastic phase transition makes the analysis of the solid Riemann problem more difficult. In this paper, one-dimensional elastic-perfectly plastic solid Riemann problem is investigated and its exact Riemann solver is proposed. Different from previous works treating the elastic and plastic phases integrally, we resolve the elastic wave and plastic wave separately to understand the complicate nonlinear waves within the solid and then assemble them together to construct the exact Riemann solver for the elastic-perfectly plastic solid. After that, the exact solid Riemann solver is associated with the fluid Riemann solver to decouple the fluid-solid multi-material interaction. Numerical tests, including gas-solid, water-solid high-speed impact problems are simulated by utilizing the modified ghost fluid method (MGFM).

An isogeometric discontinuous Galerkin method for Euler equations is proposed. It integrates the idea of isogeometric analysis with the discontinuous Galerkin framework by constructing each element through the knots insertion and degree elevation techniques in non-uniform rational B-splines. This leads to the solution inherently shares the same function space as the non-uniform rational B-splines representation, and results in that the curved boundaries as well as the interfaces between neighboring elements are naturally and exactly resolved. Additionally, the computational cost is reduced in contrast to that of structured grid generation. Numerical tests demonstrate that the presented method can be high order of accuracy and flexible in handling curved geometry. Copyright © 2016 John Wiley & Sons, Ltd.

A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible NavierStokes equations on arbitrary grids in the framework of DG methods. The DDG method, originally introduced for scalar diffusion problems on structured grids, is extended to discretize viscous and heat fluxes in the NavierStokes equations. Two approaches of implementing the DDG method to compute numerical diffusive fluxes for the NavierStokes equations are presented: one is based on the conservative variables, and the other is based on the primitive variables. The importance of the characteristic cell size used in the DDG formulation on unstructured grids is examined. The numerical fluxes on the boundary by the DDG method are discussed. A number of test cases are presented to assess the performance of the DDG method for solving the compressible NavierStokes equations. Based on our numerical results, we observe that DDG method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used BR2 method at a significantly reduced cost, clearly demonstrating that the DDG method provides an attractive alternative for solving the compressible NavierStokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost.

A class of reconstructed discontinuous Galerkin methods is described for solving compressible flow problems on arbitrary grids. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via a so-called in-cell reconstruction process. The devised in-cell reconstruction is aimed to augment the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. These three reconstructed discontinuous Galerkin methods are used to compute a variety of compressible flow problems on arbitrary meshes to assess their accuracy. The numerical experiments demonstrate that all three reconstructed discontinuous Galerkin methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy, efficiency, and robustness I. Abstract The discontinuous Galerkin methods 1-25 (DGM) have recently become popular for the solution of systems of conservation laws. Nowadays, they are widely used in computational fluid dynamics, computational acoustics, and computational electromagnetics. The discontinuous Galerkin methods combine two advantageous features commonly associated to finite element and finite volume methods. As in classical finite element methods, accuracy is obtained by means of high-order polynomial approximation within an element rather than by wide stencils as in the case of finite volume methods. The physics of wave propagation is, however, accounted for by solving the Riemann problems that arise from the discontinuous representation of the solution at element interfaces. In this respect, the methods are therefore similar to finite volume methods. The discontinuous Galerkin methods have many attractive features:1) They have several useful mathematical properties with respect to conservation, stability, and convergence; 2) The method can be easily extended to higher-order (>2 nd

In this paper, a general high-order multi-domain hybrid DG/WENO-FD method, which couples a p th -order (p � 3) DG method and a q th -order (q � 3) WENO-FD scheme, is developed. There are two possible coupling approaches at the domain interface, one is non-conservative, the other is conservative. The non-conservative coupling approach can preserve optimal order of accuracy and the local conservative error is proved to be upmost third order. As for the conservative coupling approach, accuracy analysis shows the forced conservation strategy at the coupling interface deteriorates the accuracy locally to firstorder accuracy at the ‘coupling cell’. A numerical experiments of numerical stability is also presented for the non-conservative and conservative coupling approaches. Several numerical results are presented to verify the theoretical analysis results and demonstrate the performance of the hybrid DG/WENO-FD solver.

The discontinuous Galerkin (DG) methods have attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. However, the DG methods have their own weakness due to the high computational costs and storage requirements. In this work, we develop a high-order hybrid reconstructed DG (rDG) method for solving the INS equations on arbitrary grids. To be specific, the inviscid term of the INS equations is discretized by applying the third-order hybrid rDG( P 1 P 2 ) method with a simplified artificial compressibility flux, while the viscous term of the INS equations is discretized by using the simple direct DG (DDG) method. A number of incompressible flow problems, in both steady and unsteady forms, for a variety flow conditions are computed to numerically assess the performance of the hybrid rDG( P 1 P 2 ) method, which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs. Furthermore, a detailed comparison of a variety of different reconstructed strategies is performed and presented. Numerical results demonstrate that the hybrid rDG( P 1 P 2 ) method outperforms the rDG( P 1 P 2 ) method based on either the original least-squares reconstruction or the Green-Gauss reconstruction for solving the INS equations.

In this paper, an adjoint-based airfoil optimization algorithm is developed based on isogeometric discontinuous Galerkin (IDG) method for compressible Euler equations. We first parameterize the airfoil by B-spline curve approximation with some control points viewed as design variables, and build the B-spline representation of the flow field with the curve to apply global refined IDG method for flow solution. With the isogeometric nature, not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables. Consequently, the gradient involved in SQP optimization algorithm is totally estimated in an accurate approach indicating that the numerical solutions and objective could be differentiable with respect to those variables. The proposed algorithm is demonstrated on RAE2822 airfoil with inviscid transonic flow.

The modified ghost fluid method (MGFM) provides a robust and efficient interface treatment for various multi-medium flow simulations and some particular fluid-structure interaction (FSI) simulations. However, this methodology for one specific class of FSI problems, where the structure is plate, remains to be developed. This work is devoted to extending the MGFM to treat compressible fluid coupled with a thin elastic plate. In order to take into account the influence of simultaneous interaction at the interface, a fluid-plate coupling system is constructed at each time step and solved approximately to predict the interfacial states. Then, ghost fluid states and plate load can be defined by utilizing the obtained interfacial states. A type of acceleration strategy in the coupling process is presented to pursue higher efficiency. Several one-dimensional examples are used to highlight the utility of thismethod over looselycoupled method and validate the acceleration techniques. Especially, this method is applied to compute the underwater explosions (UNDEX) near thin elastic plates. Evolution of strong shock impacting on the thin elastic plate and dynamic response of the plate are investigated. Numerical results disclose that this methodology for treatment of the fluid-plate coupling indeed works conveniently and accurately for different structural flexibilities and is capable of efficiently simulating the processes of UNDEX with the employment of the acceleration strategy.

In this paper, we develop two versions of multidomain hybrid methods by combining Runge--Kutta discontinuous Galerkin (RKDG) methods and weighted essentially nonoscillatory (WENO) schemes. One is a conservative version based on a third order RKDG method and a fifth order finite volume WENO (WENO-FV) scheme, the other is a nonconservative version based on a third order RKDG method and a fifth order finite difference WENO (WENO-FD) scheme. At the artificial interface of coupling RKDG and WENO, special treatments are used to tackle with discontinuities such as shock waves and preserve high order accuracy for smooth solution as well. We extend the nonconservative multidomain hybrid RKDG and WENO-FD (RKDG+WENO-FD) method to one- and two-dimensional systems of conservation laws for consideration of computational efficiency. Theoretical analysis shows the hybrid RKDG+WENO-FD method has high order accuracy for smooth solution, and numerical results also demonstrate that the hybrid solver is robust for slow and st...

A new reconstructed Discontinuous Galerkin (rDG) method based on a hybrid leastsquares recovery and reconstruction, named P1P2(HLSr), is developed for solving the compressible Euler and Navier-Stokes equations on arbitrary grids. Unlike the recoverybased DG method where a quadratic polynomial solution is recovered from the underlying linear DG solution and the reconstruction-based DG method where a quadratic polynomial solution is reconstructed from the underlying linear DG solution, the new hybrid rDG method obtains a quadratic polynomial solution from the underlying linear solution by using a hybrid recovery and reconstruction strategy. The developed hybrid rDG method combines the simplicity of the reconstruction-based DG method and the accuracy of the recovery-based DG method, and has the desired property of 2-exactness. A number of test cases for a variety of flow problems are presented to assess the performance of the new P1P2(HLSr) method. Numerical experiments demonstrate that this hybrid rDG method is able to achieve the designed optimal 3rd order of accuracy for both inviscid and viscous flows and outperform the rDG methods based on either Green-Gauss or least-squares reconstruction.

In this paper, we develop the Runge–Kutta discontinuous Galerkin (RKDG) methods with the finite volume weighted essentially non-oscillatory (WENO) reconstruction as limiters to solve for the unsteady cavitating flow under the employment of the isentropic one-fluid model. To treat the cavitating flow and suppress the possible spurious oscillations in the vicinity of the cavitation boundary, the TVB limiter is used as an indicator to detect the “troubled cells” and hence take the advantage of utilizing the WENO reconstruction for the freedoms of the RKDG methods. Numerical results are provided to illustrate the viability of these procedures.

In this work, the modified ghost fluid method is developed to deal with 2D compressible fluid interacting with elastic solid in an Euler-Lagrange coupled system. In applying the modified Ghost Fluid Method to treat the fluid-elastic solid coupling, the Navier equations for elastic solid are cast into a system similar to the Euler equations but in Lagrangian coordinates. Furthermore, to take into account the influence of material deformation and nonlinear wave interaction at the interface, an Euler-Lagrange Riemann problem is constructed and solved approximately along the normal direction of the interface to predict the interfacial status and then define the ghost fluid and ghost solid states. Numerical tests are presented to verify the resultant method.

Since the (original) ghost fluid method (OGFM) was proposed by Fedkiw et al. in 1999 [5], a series of other GFM-based methods such as the gas-water version GFM (GWGFM), the modified GFM (MGFM) and the real GFM (RGFM) have been developed subsequently. Systematic analysis, however, has yet to be carried out for the various GFMs on their accuracies and conservation errors. In this paper, we develop a technique to rigorously analyze the accuracies and conservation errors of these different GFMs when applied to the multi-medium Riemann problem with a general equation of state (EOS). By analyzing and comparing the interfacial state provided by each GFM to the exact one of the original multi-medium Riemann problem, we show that the accuracy of interfacial treatment can achieve ''third-order accuracy'' in the sense of comparing to the exact solution of the original mutli-medium Riemann problem for the MGFM and the RGFM, while it is of at most ''first-order accuracy'' for the OGFM and the GWGFM when the interface approach is actually near in balance. Similar conclusions are also obtained in association with the local conservation errors. A special test method is exploited to validate these theoretical conclusions from the numerical viewpoint.

The study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view. Spherical explosion and implosion in air, water and other media are well-known problems in application. Typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center, in addition to the resolution of classical discontinuities (shocks and contact discontinuities). In the present paper we present the implementation of direct generalized Riemann problem (GRP) scheme to resolve this issue. The scheme is obtained directly by the time integration of the fluid flows. Our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center. Together with the vanishing of the momentum, we obtain new numerical boundary conditions at the center, which are then incorporated into the GRP scheme. The main ingredient is the passage from the Cartesian coordinates to the radially symmetric coordinates.

An adaptive ghost fluid finite volume method is developed for one- and two-dimensional compressible multi-medium flows in this work. It couples the real ghost fluid method (GFM) [C.W. Wang, T.G. Liu, B.C. Khoo, A real-ghost fluid method for the simulation of multi-medium compressible flow, SIAM J. Sci. Comput. 28 (2006) 278-302] and the adaptive moving mesh method [H.Z. Tang, T. Tang. Moving mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487-515; H.Z. Tang, T. Tang, P.W. Zhang, An adaptive mesh redistribution method for non-linear Hamilton-Jacobi equations in two- and three-dimensions, J. Comput. Phys. 188 (2003) 543-572], and thus combines their advantages. This work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the GFM around the material interface and other discontinuities. Besides the improvement of flow field resolution, the adaptive GFM also largely increases the computational efficiency. Several numerical experiments are conducted to demonstrate robustness and efficiency of the current method. They include several 1D and 2D gas-water flow 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 for solutions with large density and pressure gradients.

In this work, the modified ghost fluid method (MGFM) [T. G. Liu, B. C. Khoo, and K. S. Yeo, J. Comput. Phys., 190 (2003), pp. 651-681] is further developed and applied to treat the compressible fluid-compressible structure coupling. To facilitate theoretical analysis, the structure is modeled as elastic-plastic material with perfect plasticity and constituted with the hydro-elasto-plastic equation of state [H. S. Tang and F. Sotiropoulos, J. Comput. Phys., 151 (1999), pp. 790-815] under strong impact. This results in the coupled compressible fluid-compressible structure system which is fully hyperbolic. To understand the effect of structure deformation on the interfacial and flow status, the compressible fluid-compressible structure Riemann problem is analyzed in the consideration of material deformation with an approximate Riemann problem solver proposed to take into account the effect of material elastic-plastic deformation. We clearly show the ghost fluid method can be applied to treat the flow-deformable structure coupling under strong impact provided that a proper Riemann problem solver is used to predict the ghost fluid states. And the resultant MGFM can work effectively and efficiently in such situations. Various examples are presented to validate and support the conclusions reached.

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

This note gives a numerical investigation for the popular high resolution conservative schemes when applied to inviscid, compressible, perfect gas flows with an initial high density ratio as well as a high pressure ratio. The results show that they work very inefficiently and may give inaccurate numerical results even over a very fine mesh when applied to such a problem. Numerical tests show that increasing the order of accuracy of the numerical schemes does not help much in improving the numerical results. How to cure this difficulty is still open.

An analysis is carried out for the ghost fluid method (GFM) based algorithm as applied to the gas-water Riemann problems, which can be construed as two single-medium GFM Riemann problems. It is found that the inability to provide correct and consistent Riemann waves in the respective real fluids by these two GFM Riemann problems may lead to inaccurate numerical results. Based on this finding, two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent Riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity. Using these two conditions to analyse some of the existing 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 original GFM or the new version GFM or both are unable to provide correct or consistent Riemann waves in one of the real fluids. Within these ranges, examples can be found such that either the original GFM or the new version GFM or both are unable to provide accurate results. The MGFM is also found to encounter difficulties when applied to nearly cavitating flow. Various examples are presented to demonstrate the conclusions obtained. The MGFM with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results.

Unlike attached cavitation, where the cavitation boundary is steady or changes relatively slowly and periodically, the cavitation such as that observed in an underwater explosion consists of a dynamically developing boundary and can evolve to a certain dimension before collapsing very violently. The development and collapse of such cavitation is sustained mainly by the pressure jump across the cavitation boundary. In this work, the focus is on developing a one-fluid model for such cavitating flows. Alter the analysis and discussion are carried out for some existing one-fluid cavitation models, such as Vacuum model, Cut-off model and Schmidt's model, a mathematically more consistent one-fluid model is then developed to study the creation, evolution and collapse of such unsteady cavitation by assuming that the cavitating flow is a homogeneous mixture of isentropic gas and liquid components. In the model, both the ambient water and the mixture of cavitating flow are taken as compressible. Besides the theoretical analysis, the present model is also tested against various problems with either exact solution, or experimental data or comparison to other existing models, and then applied to a 3D underwater problem in a cylinder.

An adaptive mesh refinement algorithm based on a continuous adjoint approach is developed. Both the primal equation and the adjoint equation are approximated with the discontinuous Galerkin (DG) method. The proposed adaptive algorithm is used in compressible Euler equations. Numerical tests are made to show the superiority of the proposed adaptive algorithm.