Solution Of Compressible Euler Equations Research Articles

Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density

We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. Various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded and the wave propagation is not at a finite speed starting initially. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit for the Navier-Stokes approximations constructed in this paper even when the total initial-energy is unbounded.

On two-dimensional steady hypersonic-limit Euler flows passing ramps and radon measure solutions of compressible Euler equations

We proposed rigorous definitions of Radon measure solutions for boundary value problems of steady compressible Euler equations which modeling hypersonic-limit inviscid flows passing two-dimensional ramps, and their interactions with still gas and pressureless jets. We proved the Newton-Busemann pressure law of drags of body in hypersonic flow, and constructed various physically interesting measure solutions with density containing Dirac measures supported on curves, also exhibited examples of blow up of certain measure solutions. This established a mathematical foundation for applications in engineering and further studies of measure solutions of compressible Euler equations.

Vanishing viscosity limit of the compressible Navier-Stokes equations with finite energy and total mass

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α=1,γ=2).

The Riemann solutions comparison for compressible Euler equations in different Chaplygin gas states

In this article, we make several comparisons of Riemann solutions of compressible Euler equations in the modified Chaplygin gas state () and classical Chaplygin gas state (). Firstly, through the analysis of characteristic method, we construct the Riemann solutions for the modified Chaplygin gas state and compare the Riemann solutions with different coefficients A. Then we obtain the reason of the occurring of delta shock waves in classical Chaplygin gas case, which is the lower bounded of 1-wave curves and the upper bounded of 3-wave curves. Moreover, by comparing the position of simple wave curves between the modified and classical Chaplygin gas state, we find that no matter how small the coefficient A is, there is an initial Riemann data, subject to that the Riemann solutions for modified Chaplygin gas state contain rarefaction waves while the Riemann solutions for classical Chaplygin gas state are delta waves, which is different from the initial expect. Finally, we present a numerical analysis consistent with the theoretical analysis.

$L^1$-Convergence to Generalized Barenblatt Solution for Compressible Euler Equations with Time-Dependent Damping

The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(\rho)=\kappa\rho^{\gamma}, \gamma>1$) with time dependent damping like $-\frac{1}{(1+t)^\lambda}\rho u$ ($0<\lambda<1$) is investigated. By introducing an elaborate iterative method and using the intensive entropy analysis, it is proved that the $L^\infty$ entropy solution of compressible Euler equations with finite initial mass converges strongly in the natural $L^1$ topology to a fundamental solution of porous media equation (PME) with time-dependent diffusion, called by generalized Barenblatt solution. It is interesting that the $L^1$ decay rate is getting faster and faster as $\lambda$ increases in $(0, \frac{\gamma}{\gamma+2}]$, while is getting slower and slower in $[ \frac{\gamma}{\gamma+2}, 1)$.

Blowup for $C^1$ solutions of compressible Euler equations with time-dependent damping

Blowup for $C^1$ solutions of compressible Euler equations with time-dependent damping

Some special self-similar solutions for a model of inviscid liquid-gas two-phase flow

In this article, we are concerned with analytical solutions for a model of inviscid liquid-gas two-phase flow. On the basis of Yuen’s works [25, 27–29] on self-similar solutions for compressible Euler equations, we present some special self-similar solutions for a model of inviscid liquid-gas two-phase flow in radial symmetry with and without rotation, and in elliptic symmetry without rotation. Some blowup phenomena and the global existence of the solutions obtained are classified.

Global expanding solutions of compressible Euler equations with small initial densities

We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of the flow.

Blowup for projected 2-dimensional [formula omitted] solutions of compressible Euler equations with Coriolis force

Compressible Euler equations with Coriolis force are the fundamental model in the study of atmospheric dynamics. In this article, we study the lifespan of the projected 2-dimensional C2 non-vacuum solutions of the compressible Euler equations with Coriolis force. We first locate the vortex equations for the compressible Euler equations with Coriolis force and then use the integration method to prove that the corresponding blowup results exist for the rotational C2 solutions with a sufficiently large initial functional, H(0)=∫R3x→⋅u→0dV.Based on these new results for the system, the presence of Coriolis forces makes the inflation conditions more difficult to be satisfied.

$$L^{1}$$ convergences and convergence rates of approximate solutions for compressible Euler equations near vacuum

In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter $$\nu $$ . The solutions $$\rho _\nu $$ and $$v_\nu $$ of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as $$\nu $$ approaches 0, the solutions $$\rho _\nu $$ and $$v_\nu $$ converge to the solutions $$\rho $$ and v, respectively, of pressureless compressible Euler equations in $$L^1$$ sense. In addition, the $$L^1$$ convergence rates of these physical quantities in terms of $$\nu $$ are also investigated. The $$L^1$$ convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of $$\partial _{x}^{i} \rho _{\nu }$$ ( $$i=0,1,2$$ ) and $$\partial _{x}^{j} v_{\nu }$$ ( $$j=0,1,2,3$$ ) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of $$\nu $$ . These theoretic results are also supported by numerical simulations.

On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls

In this paper, we are concerned with the global smooth solution problem for 3-D compressible isentropic Euler equations with vanishing density in an infinitely expanding ball. It is well-known that the classical solution of compressible Euler equations generally forms the shock as well as blows up in finite time due to the compression of gases. However, for the rarefactive gases, it is expected that the compressible Euler equations will admit global smooth solutions. We now focus on the movement of compressible gases in an infinitely expanding ball. Because of the conservation of mass, the fluid in the expanding ball becomes rarefied meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. We will confirm this interesting phenomenon from the mathematical point of view. Through constructing some anisotropy weighted Sobolev spaces, and by carrying out the new observations and involved analysis on the radial speed and angular speeds together with the divergence and rotations of velocity, the uniform weighted estimates on sound speed and velocity are established. From this, the pointwise time-decay estimate of sound speed is obtained, and the smooth gas fluids without vacuum are shown to exist globally.

Global and blow-up solutions for compressible Euler equations with time-dependent damping

This paper deals with the Cauchy problem for the compressible Euler equations with time-dependent damping, where the time-vanishing damping in the form of μ(1+t)λ makes some fantastic variety of the dynamic system. For 0<λ<1 and μ>0, or λ=1 but μ>2, the solutions are proved to exist globally in time, when the derivatives of the initial data are small, but the initial data themselves can be arbitrarily large. This is the so-called challenging case of global solutions with large initial data; while, when the initial Riemann invariants are monotonic and their derivatives with absolute value are large at least at one point, then the solutions are still bounded, but their derivatives will blow up at finite time, somewhat like the singularity formed by shock waves. For λ>1 and μ>0, or λ=1 but 0<μ≤1, the derivatives of solutions will blow up even for all initial data, including the interesting case of blow-up solutions with small initial data. Here the initial Riemann invariants are monotonic. In fact, such a blow-up phenomenon is determined by the mechanism of the dynamic system itself. In order to prove the global existence of solutions with large initial data, we introduce a new energy functional related to the Riemann invariants, which crucially enables us to build up the maximum principle for the corresponding Riemann invariants, and the uniform boundedness for the local solutions. Finally, numerical simulations in different cases are carried out, which further confirm our theoretical results.

Blowup for Projected $$\mathbf {2}$$-Dimensional Rotational $${\mathbf {C}}^{2}$$ Solutions of Compressible Euler Equations

Blowup for Projected $$\mathbf {2}$$-Dimensional Rotational $${\mathbf {C}}^{2}$$ Solutions of Compressible Euler Equations

On the Motion of a Compressible Gravity Water Wave with Vorticity

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in Lindblad and Luo (Commun Pure Appl Math, 2008) to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed $$\kappa $$ . As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Lindblad and Luo (2008) to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.

All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler case. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. (J Sci Comput 68:975–1001, 2016). Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performance of our scheme in both compressible and incompressible regimes.

A Priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit

In this paper, we prove a new type of energy estimate for the compressible Euler equations with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed k. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin to when you have a free boundary. In the incompressible case our energies reduce to those in Christodoulou and Lindblad, and our proof in particular gives a simplified proof of their estimates with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results, our result leaves open the possibility of long‐time existence also for slightly compressible liquids with a free surface.© 2017 Wiley Periodicals, Inc.

Hybrid monotonicity-preserving piecewise parabolic method for compressible Euler equations

In this article, we present a high-accuracy high-resolution hybrid scheme with strong robustness for solutions of compressible Euler equations, particularly those relating to strong discontinuity. The fifth-order monotonicity-preserving (MP5) scheme of Suresh and Huynh has high accuracy; however, its robustness is relatively weak. To improve the robustness and resolution, the MP5 scheme is conjugated with a fourth-order piecewise parabolic method (PPM) in a complementary approach. An adaptive constraint is used to detect which scheme is applied at the current position to guarantee the strong robustness and high accuracy. Through numerical experiments, our hybrid scheme demonstrates a stronger robustness and higher resolution than the traditional scheme and the hybrid scheme MP5-R by He without loss of accuracy.

Healing of the Carbuncle Phenomenon for AUSMDV Scheme on Triangular Grids

Abstract The carbuncle phenomenon, commonly occurring in solutions of compressible Euler equations, is a numerical instability associated with shock-induced anomalies. It is associated with several shock-capturing finite-volume methods designed to preserve the contact discontinuities. Due to the lack of theoretical knowledge of the carbuncle phenomenon, it is not known which numerical scheme is affected or under what circumstances that the phenomenon occur. The objective of this article is to study the numerical instability of advection upstream splitting method (AUSM) family schemes so called the AUSMD, AUSMV and AUSMDV schemes in two-dimensional structured triangular grids by examining the shock-induced anomalies produced by these original schemes in different test cases. A multidimensional dissipation technique is proposed for these schemes. The evolution of perturbations is also studied by means of a linearized discrete analysis to the odd–even decoupling problem. The recursive equations show that the AUSMDV-family schemes with the dissipation technique are less sensitive to these anomalies than the original schemes. Finally, the dissipation technique is extended to the second-order schemes and tested by several test cases.

On Dependence of Structure of Numerical Solutions of Compressible Euler Equations on Random Noises

On Dependence of Structure of Numerical Solutions of Compressible Euler Equations on Random Noises

Numerical Solution of Compressible Euler Equations by High Order Nodal Discontinuous Galerkin Method

In this paper we present a robust, high order method for numerical solution of compressible Euler Equations of the gas dynamics. Euler equations are hyperbolic in nature. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method combines mainly two key ideas which are based on the finite volume and finite element methods. In this method, we employ Discontinuous Galerkin (DG) technique for finite element space discretization by discontinuous approximations. Whereas, for temporal discretization, we used explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. We used filter to remove spurious oscillations near the shock waves. Numerical predictions for Shock tube problem (SOD) are presented and compared with exact solution at different polynomial order and mesh sizes. Results show the suitability of DG method for modeling gas dynamics equations and effectiveness of high order approximations.