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.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

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.

Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in One-Dimensional Domains

We prove that as the viscosity and heat-conductivity coefficients tend to zero, respectively, the global solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

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.

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section.Under the conical assumption,the three-dimensional(3-D)steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles.Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane,we construct a classical conical-supersonic solution near the curve under some reasonable assumptions.To overcome the difficulty caused by the parabolic degeneracy,we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane.The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully.We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Blowup of Solutions to the Non-Isentropic Compressible Euler Equations with Time-Dependent Damping and Vacuum

This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data is assumed to be radially symmetric and the initial density contains vacuum, we obtain that classical solution, especially the density, will blow up on finite time. The results also reveal that damping can really delay the singularity formation.

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density- dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel~ and the continuation argument.

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series ex- pansion technique is proposed, and a series of reduced one-dimensional （1D） equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degen- erated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic be- haviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.

NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations

In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.

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.

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.

能量泛函及Euler-Lagrange方程在图像降噪中的应用研究

研究了自适应分数阶偏微分方程修正模型的能量泛函及Euler-Lagrange方程。首先,定义了自适应分数阶偏微分方程修正模型的能量泛函,其中包含未知函数和拉格朗日乘子的集合。然后,通过求解能量泛函的极值方程,推导出了Euler-Lagrange方程。最后,讨论了Euler-Lagrange方程在自适应分数阶偏微分方程修正模型中的应用。

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.

TWO-DIMENSIONAL RIEMANN PROBLEMS:FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.

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.