THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

MESH FREE ADAPTIVE ALGORITHM FOR SOLVING EULER EQUATIONS ON STRUCTURED GRID POINTS

A complete mesh free adaptive algorithm （MFAA）, with solution adaptation and geometric adaptation, is developed to improve the resolution of flow features and to replace traditional global refinement techniques in structured grids. Unnecessary redundant points and elements are avoided by using the mesh free local clouds refinement technology in shock influencing regions and regions near large curvature places on the boundary. Inviscid compressible flows over NACA0012 and RAE2822 airfoils are computed. Finally numerical results validate the accuracy of the above method.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

UNCONDITIONAL STABLE SOLUTIONS OF THE EULER EQUATIONS FOR TWO－AND THREE－D WINGS IN ARBITRARY MOTION

The work presented here shows the unsteady inviscid results obtained for the twoand three-dimensional wings which are in rigid and flexible osciliations.The results are generated by a finite volume Euler method. It is based on theRunge- Kutta time stepping scheme developed by Jameson et al.. To increase the timestep which is limited by the stability of Runge-Kutta scheme, the implicit residualsmoothing which is modified by using variable coefficients io prerent the loss of flowphysics for the unsteady flows is engaged in the calculations. With this unconditionalstable solver the unsteady flws about the wings in arbitrary motion can be receivedefficiently.The two- and three-dimensional rectangular wings which are in rigid andflexible pitching oscillations in the transonic flow are invesigated here, some of thecomputational results are compared with the experimental data. The influence of thereduced frequency for the two kinds of the wings are researched. All the results givenin this work are reasonable.

A Gridless-Finite Volume Hybrid Algorithm for Euler Equations

A fast hybrid algorithm based on gridless method coupled with finite volume method （FVM） is developed for the solution to Euler equations. Compared with pure gridless method, the efficiency of the hybrid algorithm is improved to the level of finite volume method for most parts of the flow filed arc covered with grid cells. Moreover, the hybrid method is flexible to deal with the configurations as clouds of points are used to cover the region adjacent to the bodies. Mirror satellites and mirror grid cells arc introduced to the interface to accomplish data communication between the different parts of the flow field. The Euler Equations arc spatially discretized with finite volume method and gridless method in mesh and clouds of points respectively, and an explicit four-stage Runge-Kutta scheme is utilized to reach the steady-state solution. Internal flows in channels and external flows over airfoils arc investigated with hybrid method, and the solutions arc comparad to those using pure finite volume method and pure gridless method. Numerical examples show that the hybrid algorithm captures the shock waves accurately, and it is as efficient as fmite volume method.

A Multigrid Block LU-SGS Algorithm for Euler Equations on Unstructured Grids

We propose an efficient and robust algorithm to solve the steady Euler equa- tions on unstructured grids.The new algorithm is a Newton-iteration method in which each iteration step is a linear multigrid method using block lower-upper symmetric Gauss-Seidel(LU-SGS)iteration as its smoother To regularize the Jacobian matrix of Newton-iteration,we adopted a local residual dependent regularization as the replace- ment of the standard time-stepping relaxation technique based on the local CFL number The proposed method can be extended to high order approximations and three spatial dimensions in a nature way.The solver was tested on a sequence of benchmark prob- lems on both quasi-uniform and local adaptive meshes.The numerical results illustrated the efficiency and robustness of our algorithm.

ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE

This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term --u/（1＋t）λpu, where λ≥ 0 and μ 〉 0 are constants. We have showed that, for all λ ≥ 0 andμ 〉 0 the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial- boundary value problem in the half space R＋^d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 〈 1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume （RKCV） discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin （RKDG） method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.

STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

In this paper, we establish the existence of four families of simple wave solu- tion for two dimensional compressible full Euler system in the self-similar plane. For the 2 × 2 quasilinear non-reducible hyperbolic system, there not necessarily exists any simple wave solution. We prove the result that there are simple wave solutions for this 4× 4 non- reducible hyperbolic system, its simple wave flow is covered by four straight characteristics λ0 =λ1,λA2, λ3 and the solutions keep constants along these lines. We also investigate the existence of simple wave solution for the isentropic relativistic hydrodynamic system in the self-similar plane.

Some Remarks on Euler Equations with Damping in Multi-Dimensions

We consider the Cauchy problem of Euler equations with damping. Based on the Green function and energy estimates of solutions, we improve the pointwise estimates and obtainL 1 estimate of solutions.

REMARKS ON THE NONLINEAR INSTABILITY OF INCOMPRESSIBLE EULER EQUATIONS

In this paper, we consider the nonlinear instability of incompressible Euler equations. If a steady density is non-monotonic, then the smooth steady state is a nonlinear instability. First, we use variational method to find a dominant eigenvalue which is important in the construction of approximate solutions, then by energy technique and analytic method, we obtain the dynamical instability result.

Self-adjusting entropy-stable scheme for compressible Euler equations

In this work,a self-adjusting entropy-stable scheme is proposed for solving compressible Euler equations.The entropy-stable scheme is constructed by combining the entropy conservative flux with a suitable diffusion operator.The entropy has to be preserved in smooth solutions and be dissipated at shocks.To achieve this,a switch function,which is based on entropy variables,is employed to make the numerical diffusion term be automatically added around discontinuities.The resulting scheme is still entropy-stable.A number of numerical experiments illustrating the robustness and accuracy of the scheme are presented.From these numerical results,we observe a remarkable gain in accuracy.

Interaction of elementary waves for relativistic Euler equations

In this paper, using the characteristic analysis method, we study the relativistic Euler equations of conservation laws in energy and momentum in special relativity. The interactions of elementary waves for the relativistic Euler equations are shown. The collision of two shocks, two centered rarefaction waves, a shock and a rarefaction wave yield corresponding ransmitted waves. The overtaking of two shocks appears a transmitted shock wave, together with a reflected centered rarefaction wave.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

Adaptive Moving Mesh Central-Upwind Schemes for Hyperbolic System of PDEs:Applications to Compressible Euler Equations and Granular Hydrodynamics

We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh.When the evolution step is complete,the grid points are redistributed according to the moving mesh differential equation.Finally,the evolved solution is projected onto the new mesh in a conservative manner.The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems.Our numerical results demonstrate that in both cases,the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.

A Short and Simple Solution of the Millennium Problem about the Navier-Stokes Equations and Similarly for the Euler Equations

This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). The proof is based on an appropriate modified extension in the viscous case of the well-known Helmholtz-Kelvin-Stokes theorem of invariance of the circulation of velocity in the Euler inviscid flows. This is essentially a 1D line density of (rotatory) momentum conservation. We discover a similar 2D surface density of (rotatory) momentum conservation. These conservations are indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity.

HIGH RESOLUTION POSITIVITY-PRESERVING DIFFERENCE SCHEMES FOR TWO DIMENSIONAL EULER EQUATIONS

A class of high resolution positivity preserving Boltzmann type difference schemes for one and two dimensional Euler equations is studied. First, the relation between Boltzmann and Euler equations is analyzed. By using a kind of special interpolation, the high resolution Boltzmann type difference scheme is constructed. Finally, numerical tests show that the schemes are effective and useful.

Improving Solution of Euler Equations by a Gas-Kinetic BGK Method

Aim.The well known JST(Jameson-Schmidt-Turkel) scheme requires the use of a dissipation term.We propose using gas-kinetic BGK(Bhatnagar-Gross-Krook) method,which is based on the more fundamental Boltzmann equation,in order to obviate the use of dissipation term and obtain,we believe,an improved solution.Section 1 deals essentially with three things:(1) as analytical solution of molecular probability density function at the cell interface has been obtained by the Boltzmann equation with BGK model,we can compute the flux term by integrating the density function in the phase space;eqs.(8) and(11) require careful attention;(2) the integrations can be expressed as the moments of Maxwellian distribution with different limits according to the analytical solution;eqs.(9) and(10) require careful attention;(3) the discrete equation by finite volume method can be solved using the time marching method.Computations are performed by the BGK method for the Sod′s shock tube problem and a two-dimensional shock reflection problem.The results are compared with those of the conventional JST scheme in Figs.1 and 2.The BGK method provides better resolution of shock waves and other features of the flow fields.

Canonical Characteristic Relation for Two Dimensional Euler Equations

We propose a new way of rewriting the two dimensional Euler equations and derive an original canonical characteristic relation based on the characteristic theory of hyperbolic systems. This relation contains the derivatives strictly along the bicharacteristic directions, and can be viewed as the 2D extension of the characteristic relation in 1D case.