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This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases. [ABSTRACT FROM AUTHOR]

In this paper, we study the steady flows to the compressible Euler-Poisson system for semiconductors with the nonzero angular velocity in a radially symmetric way in an annulus. The main purpose here is to elucidate the effect of the angular velocity in the structure of the steady flows. We show the well-posedness of all kinds of types of radially symmetric spiral flows including radial subsonic/supersonic/transonic flows, and further give a specific classification of the flow patterns under the assumption of various boundary conditions at the inner and the outer circle. Additionally, different from the purely radial case, the uniqueness of radial subsonic flow can not be obtained due to the nonlocal effect caused by the angular velocity, consequently we prove the uniqueness of the radial subsonic solution in the case without the semiconductor effect or with a small current assumption. Moreover, some new patterns of spiral flows with or without shock are observed, such as a smooth transonic flow and a supersonic-supersonic shock flow for a large relaxation time parameter. [ABSTRACT FROM AUTHOR]

The inviscid limit for the two-dimensional compressible viscoelastic equations in the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. However, when the deformation tensor effect is taken into account, our results show that the deformation tensor plays an important role in the vanishing viscosity process and can surprisingly prevent the formation of strong boundary layers. As a result we are able to justify the inviscid limit of solutions for the compressible viscous flows under the no-slip boundary condition governed by the viscoelastic equations, based on the uniform conormal regularity estimates achieved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the one-dimensional isentropic compressible Euler equations with linear damping β (t , x) ρ u in a bounded domain, which can be used to describe the process of compressible flows through a porous medium. And the model is imposed a dissipative subsonic time-periodic boundary condition. Our main results reveal that the time-periodic boundary can trigger a unique subsonic time-periodic smooth solution which is stable under small perturbations on initial data. Moreover, the time-periodic solution possesses higher regularity and stability provided a higher regular boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and large time behavior of the classical solution for the non-isentropic compressible Navier-Stokes-Allen-Cahn (NSAC) system describing the mixture motion of a two-phase immiscible viscous compressible heat-conducting flow in R 3. We first establish the global well-posedness of the Cauchy problem to the non-isentropic compressible NSAC system when the initial data is a small regular perturbation near the constant state. Then by a delicate spectral analysis and the damping structure for the perturbation equation of the phase field, we prove that the phase field decays to the phase separation state at an exponential rate while the density, velocity and the temperature converge to the constant state at the optimal time rate (1 + t) − 3 / 4 in L 2 norm. Furthermore, for well-chosen initial data, we also establish the lower bounds of time decay rate. [ABSTRACT FROM AUTHOR]

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this work, a bound- and positivity-preserving quasi-conservative discontinuous Galerkin (DG) method is proposed for the γ -based model of compressible two-medium flows. The contribution of this paper mainly includes three parts. On one hand, the DG method with the extended Harten-Lax-van Leer contact flux is proposed to solve the γ -based model, and satisfies the equilibrium-preserving property which preserves uniform velocity and pressure fields at an isolated material interface. On the other hand, an affine-invariant weighted essentially non-oscillatory (Ai-WENO) limiter is adopted to suppress oscillations near the discontinuities. The limiter with the Ai-WENO reconstruction method to the conservative variables not only is able to maintain the equilibrium property, but also generates sharper results around the locations of shock waves in contrast to that applying to the primitive variables. Last but not least, a flux-based bound- and positivity-preserving limiting strategy is introduced and analyzed, which preserves the physical bounds for auxiliary variables in the non-conservative governing equations, and the positivity for density and internal energy. Extensive numerical experiments in both one and two space dimensions show that the proposed method performs well in simulating compressible two-medium flows with high-order accuracy, equilibrium-preserving and bound-preserving properties. [ABSTRACT FROM AUTHOR]

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows. • To accelerate the convergence for steady flows, the implicit non-compact and compact HGKSs are developed for compressible flows. • For non-compact HGKS, the third-order WENO method is adopted, and the GMRES method with Jacobian matrix is used for temporal evolution. • For the compact HGKS, the third-order HWENO method is used, a simple strategy is used to update the cell averaged gradient with GMRES method. • To accelerate the computation, the current schemes are implemented with GPU. [ABSTRACT FROM AUTHOR]

An artificial viscosity-based shock capturing scheme is extended to the context of high-order triangular Spectral Difference method for solving gas dynamics problems featuring discontinuities and shock waves. The equations are regularised thanks to an artificial diffusivity method combined with a shock sensor based on dilatation and vorticity fields valid for any polynomial order of approximation. The methodological novelty of this paper is an L 2 projection based iterative Restriction Prolongation filtering operation introduced for high-order triangular elements. This filtering operation applied to the artificial bulk viscosity and thermal conductivity stabilises the simulations and improves the quality of the shock capturing approach. The methodology is validated on various canonical relevant compressible test cases. Numerical results demonstrate the performance and the flexibility of the triangular high-order Spectral Difference method for solving simultaneously compressible flows and shock waves. [ABSTRACT FROM AUTHOR]

This paper presents a model for simulating a two-phase flow involving cavitation and condensation, focusing on its application to turbopump flows. The model is derived by extending Kapila's approach, which incorporates thermal and free energy relaxation terms, to compressible multiphase flow equations in a rotating reference frame. The numerical method employed is a finite volume approach based on the Godunov scheme, ensuring pressure equilibrium and total energy conservation through a relaxation step. The model utilizes separate Equations of State (EOS) for the liquid and vapor phases, with the experimental saturation curve, expressed as p sat (T) , being the only parameter needed for cavitation modeling. To broaden its applicability beyond axial pumps, a specific Riemann solver is developed to couple rotating and non-rotating regions within the flow domain. The thermodynamic phase change process is validated using simple cases, and the model is further utilized to simulate water flows in a turbopump inducer. The characteristic curve of the pump is obtained in the non-cavitating regime for several mass flow rates conditions, and the model is successfully assessed in severe cavitation conditions, demonstrating the performance breakdown caused by vapor pockets. The modeling also provides detailed flow descriptions, including analysis of vapor pockets and their impact on blade pressure loading. • Modeling multiphase flow in rotating reference frame. • A new Riemann problem to connect fixed and rotating flow regions. • Phase change modeling by a relaxation method from pure phase to mixture. • Cavitating flow in turbopumps without parameter for the growth of the cavitation pockets. [ABSTRACT FROM AUTHOR]

This paper extends the gradient-based reconstruction approach of Chamarthi [1] to genuine high-order accuracy for inviscid test cases involving smooth flows. A seventh-order accurate scheme is derived using the same stencil as of the explicit fourth-order scheme proposed in Ref. [1] , which also has low dissipation properties. The proposed method is seventh-order accurate under the assumption that the variables at the cell centres are point values. A problem-independent discontinuity detector is used to obtain high-order accuracy. Accordingly, primitive or conservative variable reconstruction is performed around regions of discontinuities, whereas smooth solution regions apply flux reconstruction. The proposed approach can still share the derivatives between the inviscid and viscous fluxes, which is the main idea behind the gradient-based reconstruction. Several standard benchmark test cases are presented. The proposed method is more efficient than the seventh-order weighted compact nonlinear scheme (WCNS) for the test cases considered in this paper. • Seventh-order accuracy using first two moments of Legendre basis. • Genuinely high-order accuracy using discontinuity detector where smooth solution regions apply flux reconstruction. • Efficient reconstruction as the gradients are shared between viscous and inviscid fluxes. [ABSTRACT FROM AUTHOR]

The paper deals with stochastic homogenization of a system modeling immiscible compressible two-phase, such as water and gas, flow in random porous media. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. We consider the behavior of compressible two-phase flow in heterogeneous reservoirs with permeability and porosity being realizations of given statistically homogeneous random fields. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments. To our best knowledge, this is the first stochastic homogenization result in the case of compressible two-phase flow in random porous media. [ABSTRACT FROM AUTHOR]

In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and outer circle. Due to the nonzero angular velocity, many new flow patterns will appear. There exists accelerating or decelerating smooth transonic flows in an annulus satisfying one side boundary conditions at the inner or outer circle with all sonic points being nonexceptional and noncharacteristically degenerate. More importantly, it is found that besides the well-known supersonic-subsonic shock in a divergent nozzle as in the case without angular velocity, there exists a supersonic-supersonic shock solution, where the downstream state may change smoothly from supersonic to subsonic. Furthermore, there exists a supersonic-sonic shock solution where the shock circle and the sonic circle coincide, which is new and interesting. [ABSTRACT FROM AUTHOR]

Abstract A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − BVD (polynomial of 4-degree and THINC function of 2-level reconstruction based on BVD algorithm), is formulated as a two-stage spatial reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P 4 T 2 − BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with two-level steepness are used as the candidate reconstruction functions. The final reconstruction function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P 4 T 2 − BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P 4 T 2 − BVD , as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales. Highlights • An innovative fifth-order shock capturing scheme is proposed. • Low-dissipation linear scheme is retrieved for smooth solution over all wave numbers. • Discontinuous and smooth solution of wide-band scales are simultaneously solved with high accuracy. • The free-mode solutions are faithfully maintained in long term computation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes. • The all-regime acoustic/transport splitting solver proposed in [16] can be modified into a flux-splitting, one-step version. • The resulting method is less computationally expensive, more memory efficient, and easier to implement than the original one. • Stability properties are derived by interpreting the scheme as a convex combination of the advection/pressure steps. • This recasting allows an easy high-order extension as the new method is a standard flux-based solver. • Following the lines of [32] , we equip our scheme with the well-balanced property for gravity source terms. [ABSTRACT FROM AUTHOR]

In this paper, a fully conservative sharp-interface method is developed for compressible multiphase flows with phase change. To enforce conservation, the coupling between two different phases is achieved via the interfacial fluxes, which are determined by solving a general Riemann problem with phase change. In addition, thermodynamical consistency is guaranteed by taking account of latent heat in the phase-change Riemann problem. This is efficiently and robustly solved by a novel approximate Riemann solver, where the complex eight-dimensional root-finding procedure required in the exact Riemann solver is simplified to solely iterating the mass flux. Compared to the existing methods, the present approximate Riemann solution consists of four waves, at which the jump conditions are imposed strictly to enforce the conservation of mass, momentum, and energy. Furthermore, for the evaluation of the mass flux, through numerical simulations and theoretical analysis, we have demonstrated that the adjacent states of a phase interface should be used to ensure numerical consistency, which, to our knowledge, has not been reported in the literature. Various numerical examples, including the phase-change Riemann problem, the oscillating droplet problem, and the shock-droplet interaction problem, are simulated to validate the present method. The numerical results are in good agreement with the exact solutions and the previous simulations. It is shown that, with the present method, the phase change effects in compressible multiphase flows are well resolved with conservation strictly preserved. • A fully conservative sharp-interface method is developed for compressible multiphase flows with phase change. • A novel approximate Riemann solver is developed to solve the two-phase phase-change Riemann problem. • The present Riemann solver employs a 4-wave model with all the jump conditions imposed strictly, which is more physical. • We ensure numerical consistency by using the adjacent states of the phase interface to compute the mass flux. [ABSTRACT FROM AUTHOR]

When solving compressible multi-material flow problems, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular idea in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. The work presented in this paper aims to accelerate the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed, including (1) a change of integration variable through rarefaction fans, (2) storing and reusing integration trajectory data, (3) step size adaptation, and (4) constructing an R-tree on the fly to generate initial guesses. The performance of these acceleration methods is assessed using four example problems in underwater explosion, laser-induced cavitation, and hypervelocity impact. These problems exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. For all the problems, the acceleration methods are able to significantly reduce the computational cost without affecting solver robustness or solution accuracy. In different cases, the solution of bimaterial Riemann problems is accelerated by 37 to 87 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 81 times. • Bimaterial Riemann problems can be constructed to compute interfacial fluxes. • Computational cost is often prohibitive, if complex equations of state are involved. • An efficient exact Riemann solver is proposed, with four algorithmic improvements. • Significant cost reduction (37-87X) is achieved for four challenging examples. • Does not introduce approximations or require offline precomputation. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed numerical scheme for a single-phase compressible flow without diffusion of a multi-component mixture in porous media with the higher-order approximation in both space and time. The mathematical model consists of Darcy velocity, transport equations for each component of a mixture, pressure equation and associated relations for physical quantities such as viscosity or equation of state. The discrete problem is obtained using a combination of the discontinuous Galerkin method for the transport equations and the mixed-hybrid finite element method for the Darcy velocity and the pressure equation. In both methods the higher-order approximation is used. The resulting nonlinear problem for concentrations is solved with the fully mass-conservative iterative IMPEC method. Experimental order of convergence analysis (EOC) and some numerical experiments of a 2D flow are carried out. • A new numerical scheme is presented for simulation of single-phase flow in a porous medium. • The second-order approximation is attained in both space and time. • The second-order approximation in both time and space is verified using the experimental order of convergence. • Several computational examples illustrate advantageous features of our approach. [ABSTRACT FROM AUTHOR]

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared. [ABSTRACT FROM AUTHOR]

In this paper, we study the free boundary problem of compressible non-isothermal liquid crystal flow in dimension one. For the vacuum interface condition, that is, the density is assumed to vanish and connect to vacuum continuously, we prove global existence and uniqueness of classical solutions to the liquid crystal system with large initial data. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the works in [35] and [37] , where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function. [ABSTRACT FROM AUTHOR]

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations as viscosity vanishes with a convergence rate ϵ 1 / 6 | ln ⁡ ϵ |. In this paper, the vanishing viscosity limit of 2D non-isentropic compressible Navier-Stokes equations is studied. In contrast to the work [17] , the convergence rate for the 2D full compressible Navier-Stokes equations is improved to ϵ 2 / 7 | ln ⁡ ϵ | 2 by choosing a different scaling argument and performing more detailed energy estimates. [ABSTRACT FROM AUTHOR]

In the previous significant works by Alt et al. (1982) [2] and Alt et al. (1985) [6] , the existence of the two-dimensional asymmetric incompressible jet flow and compressible subsonic jet flow were established. However, the uniqueness of the asymmetric jet flow has remained an unsolved open problem. In this paper, we will establish the uniqueness of asymmetric jet flow not only for incompressible case but also for compressible subsonic case. Furthermore, as a byproduct, some properties of the jet flow are obtained, such as the location estimate and asymptotic deflection angle estimate of the jet in far field. Finally, a comparison theorem will be introduced to compressible subsonic jet flow. [ABSTRACT FROM AUTHOR]

This paper presents under-resolved simulations of compressible flows in complex structures like buildings. The mean pressure field is the primary focus as it is of high importance in many cases, such as the overpressure generated by the detonation of an explosive charge. In such high-risk situations, the mean pressure in the different building rooms has to be determined as quickly as possible, in order to help first responders evaluating the risk level of entering the building right after an explosion for instance. Conventional computational fluid dynamics methods are obviously able to deliver those requested pressure fields. However, they also require a long and tedious meshing process due to the presence of small openings in the computational domain (doors, windows or stairwells). In turn, these small openings need spatial and temporal resolution having significant consequences on computation time. But when dealing with pressing situations, a long pre-processing is not acceptable. The present paper proposes a two-step solution method. First, a very rough mesh is constructed which does not consider small openings. The mesh is constructed in an easy and particular way such that simplifications can be made when computing the fluid flux through these openings. Second, a specific Riemann solver dealing with geometric discontinuities is developed to estimate the fluid flow. The Riemann solver takes advantage of the simplifications provided by the meshing method, and accounts for both unchoked and choked flows through openings. The overall method then consists of both a specific meshing process that requires very little pre-processing, which considerably reduces the time needed to produce a complete simulation, and a Riemann solver. The proposed method is named MUZO in reference to its "MUlti-ZOne" flow solver. The MUZO solver is validated against resolved computations and provides accurate simulated mean pressure fields within a realistic building in a few seconds. • 3D simulations of compressible flow discharge in buildings are addressed through a very fast numerical framework. • Only 2D footprints without geometric details (doors, windows, etc.) are required to generate 3D meshes. • The previously omitted geometric restrictions are considered through a specific Riemann solver. [ABSTRACT FROM AUTHOR]

In this paper, we justify of the low Mach number limit of the steady irrotational Euler flow through infinite multidimensional largely-open nozzles. Firstly, the existence and uniqueness of the incompressible flow are proved. Then, the uniform estimates on the compressibility parameter ε , which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is in the Hölder space and is unique. Moreover, the convergence rate is of order ε 2 including the pressure. And, for the compressible case, the extra force influences the asymptotic behaver to the open end, while the effect vanishes in the limiting process from compressible flows to the incompressible ones. Also, for the compressible flows with general conservative forces, the well-posedness of subsonic solution are given, which leads to the existence of subsonic-sonic limit solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove a time dependent lower bound on density in the optimal order O (1 / (1 + t)) for the general smooth nonisentropic flow of compressible Euler equations. [ABSTRACT FROM AUTHOR]

• We present an ALE-RKDG method with ALE, RKDG, adaptive triangular mesh. • We prove that the material derivatives of basis functions are zero. • The ALE-RKDG scheme is proved to satisfy Geometric Conservation Law. In this paper, we present a high-order arbitrary Lagrangian Eulerian (ALE) method for compressible single-material flow on adaptive moving unstructured meshes. The discretization of system is implemented by Runge-Kutta Discontinuous Galerkin (RKDG) method. The vertex velocity is given by the approach of mesh movement from G. Chen et al. (2008) [8]. The new mesh can be automatically redistributed and concentrated on the regions with large gradient value of the variables. A HWENO reconstruction is used to eliminate false oscillations which maintains compactness of DG methods. In addition, we prove the property that the material derivatives of the Lagrangian basis functions are equal to zero with which the scheme is shown to satisfy the geometric conservation law (GCL). Furthermore, the scheme is conservative for mass, momentum and total energy. Numerical examples are presented to illustrate the good performance and high-order accuracy of the scheme. [ABSTRACT FROM AUTHOR]

The discrete-equations method (DEM) [1] provides a universal approach to solve multi-phase-flow equations as it combines the solutions of pairwise Riemann problems. Although very robust, the original DEM with piecewise-constant volume fractions suffers from strong diffusion preventing accurate interface capturing. High-order interface reconstruction, however, introduces a restrictive time-step limit. This paper presents RDEMIC, a robust extension of DEM for accurate interface capturing on Cartesian meshes. By a modified partitioning of the Riemann solutions and a specific combination of fluxes and non-conservative terms, the time-step restriction is effectively prevented, which is critical for making the method practically applicable. Moreover, the accuracy of interface and shock-wave propagation is maintained. RDEMIC is not limited to two-phase flow but defined for an arbitrary number of phases. The method is combined with a THINC scheme [2] to reconstruct volume fractions. The reconstruction is enhanced by a positivity-preserving averaging procedure, which is consistent with the underlying multi-stage Runge–Kutta scheme of the flow solver. The resulting scheme consisting of RDEMIC and the positivity-preserving THINC reconstruction is very robust and captures the interface with high accuracy. We demonstrate its performance for various cases of shock-interface interactions, which show very good agreement with reference results from literature. • Robust high-resolution numerical model and algorithm for compressible multi-phase flow with arbitrary number of phases. • No further time-step restrictions implied by high-order interface reconstruction. • Accurate capturing and advection of multi-phase interfaces with positivity preservation. • No numerical artifacts for wave propagation across interfaces. [ABSTRACT FROM AUTHOR]

Aircraft aerodynamic design often involves evaluating flow conditions that span low subsonic to transonic or even supersonic Mach numbers. Compressible flow solvers are a natural choice for such design problems, but these solvers suffer from reduced accuracy and efficiency at low Mach numbers. In addition, simulations with supersonic conditions can be challenging to converge because of large gradients in the flow field. This paper presents three contributions to address these issues in the context of an approximate Newton–Krylov solver for the Reynolds-averaged Navier–Stokes equations. First, we propose a method for scaling the artificial dissipation terms in the Jameson–Schmidt–Turkel scheme to improve its accuracy at low Mach numbers while retaining the simplicity of the original scalar dissipation formulation. Second, we show that characteristic time-stepping combined with an approximate Newton method can accelerate convergence for low Mach number flows by reducing the stiffness in the linear system for each Newton iteration. Third, we introduce a dissipation-based continuation method for flows with shocks that improves robustness and accelerates convergence without sacrificing accuracy. These methods can make compressible flow solvers more accurate and efficient across low and high Mach number regimes. • A simple change to a scalar dissipation scheme improves accuracy at low Mach numbers. • Characteristic time-stepping reduces linear system stiffness for Newton-type solvers. • Scaling artificial dissipation is needed to achieve speedups at low Mach numbers. • Dissipation-based continuation improves robustness for flows with shocks. [ABSTRACT FROM AUTHOR]

In this paper, a new type of hybrid weighted essentially non-oscillatory (WENO) schemes with third-order and fourth-order accuracy is developed to simulate compressible flow problems on unstructured meshes. First, an improved discontinuity sensor is designed on unstructured meshes, which can automatically and accurately identify the troubled cells without manually adjusting the parameters related to the problem. Second, a hybrid unequal-sized WENO (US-WENO) scheme with different accuracy is developed based on the new hybridization strategy and the existing US-WENO scheme to reduce the computational overhead caused by the computation of nonlinear weights and local characteristic decompositions in WENO procedures. Finally, some benchmark examples are employed to verify the performance of the proposed method in terms of numerical dissipation, shock capture ability, and computational efficiency. Numerical experiments show that the proposed hybrid US-WENO schemes can inherit the excellent characteristics of the original US-WENO schemes while improving computational efficiency. • An improved, simple and artificial parameter free discontinuity sensor is presented on triangular meshes. • An efficient hybrid unequal-sized WENO scheme is designed to solve compressible problems on triangular meshes. • These new hybrid WENO schemes are more efficient than the existing unequal-sized WENO scheme. [ABSTRACT FROM AUTHOR]

This paper considers a coupling of slightly compressible Darcy-Brinkman-transport problem in fractured media with higher Reynolds numbers, involving Brinkman flow in the fractures with advection-diffusion transport in the whole considered media. A new two-layer reduced coupled model is introduced by treating the fracture as hyperplane. The finite difference method with second-order backward differentiation formula and modified upwind scheme by using the fewest nodal points is constructed to solve the new reduced coupled model on staggered nonuniform grids. Based on error derivation of the coupling term, the uniqueness and existence of solutions and second-order convergence rate of numerical method are derived in the mixed forms under the mass conservative transmission and Beavers-Joseph-Saffman conditions. Some experiments are provided to testify the accuracy and efficiency of the numerical method. The numerical analysis of new two-layer reduced model is illustrated to show the behavior of fluid flow and solute transport in the media with intersected, embedded and L-shaped fractures. [ABSTRACT FROM AUTHOR]

This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary can effectively reduce the dissipation in numerical flux. Differently from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, the BVD strategy presented in this paper switches between a high-order polynomial and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. This work provides a simple and accurate alternative of great practical significance to the current high-order Godunov paradigm which overly pursues the smoothness within mesh cells under the questionable premiss that discontinuities only appear at cell interfaces. [ABSTRACT FROM AUTHOR]

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure. • A positivity-preserving high order accurate finite difference WENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • WENO reconstruction is performed on variables from a positivity-preserving convection diffusion flux splitting. • The method is robust and suitable for solving the compressible flow with low density and low pressure. [ABSTRACT FROM AUTHOR]

This is the first of two companion papers that introduce a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. In this paper, we develop and analyze a novel first–order finite volume scheme that is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The scheme is constructed by regularizing the Navier-Stokes equations by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The key distinctive property of the proposed scheme is that it is proven to satisfy the discrete entropy inequality and to guarantee the pointwise positivity of density, temperature, pressure, and internal energy for 1–D compressible viscous flows, while using high–order spectral collocation summation-by-parts operators for discretizing the Navier–Stokes viscous terms. To eliminate time step stiffness caused by the high-order approximation of the viscous terms, we develop two new limiters that bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results demonstrating positivity-preserving and discontinuity-capturing capabilities of the new scheme are presented for viscous and inviscid flows with nearly vacuum regions and very strong shocks and contact discontinuities. • Novel first-order artificial dissipation based on the Brenner diffusion operator. • New 1st-order positivity-preserving L 2 stable scheme for 1D Navier-Stokes equations. • New positivity-preserving entropy stable velocity and temperature limiters. [ABSTRACT FROM AUTHOR]

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L 2 -stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes. • A new physics-based artificial viscosity built as a function of entropy residual. • A novel high-order entropy stable pointwise positivity-preserving flux limiter. • New high-order positivity-preserving L 2 stable schemes for Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we present a pairwise-relaxing meshless (PRM) method for solving the Euler equations of compressible flows within the Arbitrary Lagrangian Eulerian (ALE) framework. Derived from the moving particle semi-implicit (MPS) method and the finite volume particle (FVP) method, the PRM approximates the derivatives from the value defined at the midpoint of each interacting particle pairs through a kernel-based formulation. Pairwise-relaxing constants are introduced to the kernels to provide degree of freedom to enforce the Taylor-series consistency condition while mass, momentum and energy are conserved exactly. An upwind high-order reconstruction scheme via a corrective procedure and variable cut-off radius is also developed for this PRM method. The HLLC approximate Riemann solver is adopted to solve Riemann problem. One and two-dimensional numerical tests are presented to demonstrate the performance of the PRM method. • Second-order accuracy and conservation property are guaranteed in the PRM. • An upwind reconstruction process via a corrective procedure is developed. • Variable cut-off radius is used. [ABSTRACT FROM AUTHOR]

Abstract The compressible Navier–Stokes–Poisson system takes the form of usual Navier–Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electrostatic potential force. In this paper, we focus on the large-time behavior of global strong solutions in the L p critical Besov spaces. Inspired by the dissipative effect arising from Poisson potential, we formulate a new regularity assumption of low frequencies and then establish the sharp time-weighted inequality, which leads to the optimal time-decay estimates of strong solutions. Indeed, we see that the decay of density is faster at the half rate than that of velocity, which is a different ingredient in comparison with the situation of compressible Navier–Stokes equations. Our proof mainly depends on tricky and non classical Besov product estimates with respect to various Sobolev embeddings. [ABSTRACT FROM AUTHOR]

Abstract For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A , is a 1-to-1, onto mapping to Σ ⁎ ? For which sets A does there exist such a mapping that respects the lexicographical ordering within A ? Both cases are types of perfect, minimal hash functions. The complexity-theoretic versions of these notions are known as compression functions and ranking functions. This paper defines and studies the recursion-theoretic versions of compression and ranking functions, and in particular studies which sets have, or lack, such functions. Thus this is a case where, in contrast to the usual direction of notion transferal, notions from complexity theory are inspiring notions, and an investigation, in computability theory. We show that the rankable and compressible sets broadly populate the 1-truth-table degrees, and we prove that every nonempty coRE cylinder is recursively compressible. [ABSTRACT FROM AUTHOR]

Abstract A systematic analysis of the discrete conservation properties of non-dissipative, central-difference approximations of the compressible Navier–Stokes equations is reported. A generalized splitting of the nonlinear convective terms is considered, and energy-preserving formulations are fully characterized by deriving a two-parameter family of split forms. Previously developed formulations reported in literature are shown to be particular members of this family; novel splittings are introduced and discussed as well. Furthermore, the conservation properties yielded by different choices for the energy equation (i.e. total and internal energy, entropy) are analyzed thoroughly. It is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family. Local conservation of primary invariants, which is a fundamental property to build high-fidelity shock-capturing methods, is also discussed in the paper. Numerical tests performed for the Taylor–Green Vortex at zero viscosity fully confirm the theoretical findings, and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results. Highlights • Energy-preserving split forms in compressible flow equations are studied. • A novel two-parameter family of energy-preserving splittings is derived. • A dynamic splitting procedure with optimal conservation properties is proposed. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Highlights • High-order accuracy in space is done via Taylor series, whereas in time it is reached through Runge–Kutta methods. • Weak consistency of staggered schemes is proved for barotropic hydrodynamics. • Weak consistency is proved for compressible hydrodynamics using an a posteriori internal energy corrector. • Wide variety of numerical test-cases illustrate the interest, the robustness and the convergence of the schemes. Abstract Staggered grids schemes, formulated in internal energy, are commonly used for CFD applications in industrial context. Here, we prove the consistency of a class of high-order Lagrange-Remap staggered schemes for solving the Euler equations in 1D and 2D on Cartesian grids. The main result of the paper is that using an a posteriori internal energy corrector, the Lagrangian schemes are proved to be conservative in mass, momentum and total energy and to be weakly consistent with the 1D Lagrangian formulation of the Euler equations. Extension in 2D is done using directional splitting methods and face-staggering. Numerical examples in both 1D and 2D illustrate the accuracy, the convergence and the robustness of the schemes. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a novel framework for hp -adaptation of high-order discontinuous finite element discretizations for compressible flow simulation. Using the sensitivities of an adjoint-based error estimate, our method seeks element size, shape, and polynomial degree distributions which minimize an adjoint-based error estimate for a specified number of degrees of freedom. This approach results in an optimized hp -mesh tailored to yield the most accurate prediction of an output quantity of interest, such as aerodynamic coefficients, at a given computational cost. The proposed approach features a reduced dependence on user-defined parameters compared to established fixed-fraction adjoint-based adaptive methods. It provides a unifying framework where adaptation decisions such as isotropic/anisotropic, h -/ p -refinement/coarsening do not only rely on local arbitrary measures of solution anisotropy and smoothness, but rather where a globally optimal distribution of degrees of freedom is sought to minimize the error in the chosen quantity of interest. Highlights • A novel framework for goal-oriented anisotropic hp -adaptation is proposed. • An hp -mesh optimization problem is solved to minimize the target functional error. • Optimal distributions of degrees of freedom are obtained via numerical optimization. • It is a unifying approach featuring reduced dependence on user-defined parameters. • Computational performance is improved compared to fixed-fraction adaptation methods. [ABSTRACT FROM AUTHOR]

Abstract This study proposes a kinetic energy and entropy preserving scheme to achieve stable and non-dissipative compressible flow simulations. The proposed scheme is built in such a way that the numerical formulations satisfy the analytical relations among the governing equations at the discrete level. The mass and momentum convective terms are recast into split convective forms. Once the mass and momentum equations are discretized, the constraints given by the analytical relations subsequently determine the formulations of the numerical fluxes solved in the total energy equation. An analysis reveals that satisfying the analytical relations at the discrete level is essential for solving the energy exchange between the kinetic energy and internal energy correctly in the total energy equation, and therefore important for entropy conservation. Taylor–Green vortex and Euler isotropic turbulence simulations at infinite Reynolds number are conducted to verify the analysis given in this paper. The proposed scheme achieves both kinetic energy conservation in the incompressible limit and entropy conservation as well as convergences of thermodynamic variable fluctuations, whereas existing kinetic energy preserving schemes fail entropy conservation and the thermodynamic variable fluctuations diverge. Highlights • Kinetic energy preservation (KEP) by split convective forms. • The total entropy is implicitly preserved at the discrete level. • The numerical fluxes satisfy the analytical relations of the governing equations. • More computationally robust scheme than existing KEP schemes for compressible flows. [ABSTRACT FROM AUTHOR]

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long-wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low-Mach-number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations can in some cases be prohibitively expensive due to the CFL time step constraint. For example, simulation of thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain. The present paper introduces a new multi-level hybrid algorithm to address these types of phenomena. In this new approach, the fully compressible Euler equations are solved on the entire domain, potentially with local refinement, while their low-Mach-number counterparts are solved on subregions of the domain with higher spatial resolution. The finest of the compressible levels communicates inhomogeneous divergence constraints to the coarsest of the low-Mach-number levels, allowing the low-Mach-number levels to retain the long-wavelength acoustics. The performance of the hybrid method is shown for a series of test cases, including results from a simulation of the aeroacoustic propagation generated from a Kelvin–Helmholtz instability in low-Mach-number mixing layers. It is demonstrated that compared to a purely compressible approach, the hybrid method allows time-steps two orders of magnitude larger at the finest level, leading to an overall reduction of the computational time by a factor of 8. [ABSTRACT FROM AUTHOR]

In this paper we develop a conservative interface-interaction method dedicated to simulating multiple compressible fluids with sharp interfaces. Numerical models for finite-volume cells cut by more than two material-interface are proposed. First, we simplify the interface interaction inside such a cell to avoid the need for explicit interface reconstruction and very complex flux calculation. Second, conservation is strictly preserved by an efficient conservation correction procedure for the cut cell. To improve robustness, a multi-material scale separation model is developed to remove consistently non-resolved interface scales. In addition, a multi-resolution method and a local time-stepping scheme are incorporated into the proposed multi-material method to speed up high-resolution simulations. Various numerical test cases, including the multi-material shock tube problem, inertial confinement fusion implosion, triple-point shock interaction and shock interaction with multi-material bubbles, show that the method is suitable for a wide range of complex compressible multi-material flows. [ABSTRACT FROM AUTHOR]

Although interpolation-based finite difference methods have shown great success in the simulation of complex configurations, the schemes currently are still inappropriate in many circumstances due to their poor spectral properties. In this paper, a class of optimized low-dissipation and low-dispersion schemes is developed with the cell-centered finite difference method (Liao, J. Comput. Phys. , 284:419–433, 2015). Firstly, the linear schemes with two free parameters which dominate the dispersion and dissipation properties independently are derived. Secondly, the two free parameters are optimized separately according to specific criteria to obtain the optimized linear schemes. Thirdly, for shock-capturing capability, the optimized non-linear schemes are acquired from the optimized linear schemes using the multi-stencil weighting technique. Meanwhile, a new series of weighting functions and smoothness indicators are derived following the idea of Hu (Hu, J. Comput. Phys. , 229:8952–8965, 2010) for the optimized non-linear schemes. Finally, several benchmark cases are validated with the developed optimized linear and non-linear schemes. In flows with shocks, the optimized non-linear schemes can give more flow details than the non-linear reference schemes without noticeable robustness loss. In the low-speed problems, the optimized linear schemes perform much better than the corresponding upwind linear schemes. Additionally, satisfactory results are observed on the wavy grid, indicating the optimized schemes with cell-centered finite difference method are promising. [ABSTRACT FROM AUTHOR]

In this paper, a semi-analytical solution is presented for the problem of modal acoustic radiation efficiency of a finite-length rotating cylinder. The cylinder is immersed in an irrotational inviscid compressible fluid. Assuming the cylinder to vibrate sinusoidally in transverse direction and rotate at constant angular velocity, the scalar equation of sound propagation is decomposed into a coupled set of time independent equations. Using Galerkin method and employing spherical eigen-functions as a complete set of solution, modal acoustic pressure and acoustic radiation efficiency of the cylinder are calculated. Numerical solutions are presented to discuss modal radiation efficiency corresponding to forward and backward circumferential waves. [ABSTRACT FROM AUTHOR]

In this paper we consider the degenerate quantum compressible Navier–Stokes equations giving rise to a variety of mathematical problems in many areas. We study the asymptotic limit for the rotational compressible Navier–Stokes equations with quantum term and the ill-prepared initial data. [ABSTRACT FROM AUTHOR]

This paper is the second part of a series of two. It follows [44] , in which the positivity-preservation property of methods solving one-dimensional Lagrangian gas dynamics equations, from first-order to high-orders of accuracy, was addressed. This article aims at extending this analysis to the two-dimensional case. This study is performed on a general first-order cell-centered finite volume formulation based on polygonal meshes defined either by straight line edges, conical edges, or any high-order curvilinear edges. Such formulation covers the numerical methods introduced in [6,32,5,41,43] . This positivity study is then extended to high-orders of accuracy. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. It is important to point out that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes. [ABSTRACT FROM AUTHOR]