The general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynamics

In this paper, we give the explicit solution to the general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynamics.     

CAUCHY PROBLEM FOR LINEARIZED SYSTEM OF TWO-DIMENSIONAL ISENTROPIC FLOW WITH AXISYMMETRICAL INITIAL DATA IN GAS DYNAMICS

§1IntroductionIn the present paper,we are interested in solving the Cauchy problem for linearizedsystem of two-dimensional isentropic flow with initial data in gas dynamicsρt+ρ0xu+yv=0,ut+p′(ρρ00)xρ=0,vt+p′(ρ0ρ0)yρ=0,(1.1)t=0:(ρ,u,v)=(ρ0(r),u0(r),v0(r)),(1.2)whereρis the density,(u,v)is the velocity,ρ0is a positive constant,p=p(ρ)is theequation of state satisfying p′(ρ0)>0,(r,θ)is the polar coordinate such thatx=rcosθ,y=rsinθ,0≤r<+∞,0≤θ≤2π…     

Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates

We prove the existence and compactness (stability) of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics, where the pressure and density are related by a γ-law, for any γ > 1. Our results considerably extend and simplify the program initiated by DiPerna and provide a complete existence proof. Our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws. © 1996 John Wiley & Sons, Inc.     

Adaptive linearized models for optimization of gas networks

We are interested in simulation and optimization of gas networks. Usually, a gas network consists of various components like compressors and valves connected by pipes. The aim is to run the network cost efficiently whereas demands of consumers have to be satisfied. This results in a complex nonlinear mixed integer problem. We address this task with methods provided by discrete optimization. Therefore, the gas dynamics in all pipes and at compressors must be described by piecewise linear constraints. We introduce an adaptive approach for the linearization process to handle the complexity on the one hand and the aimed accuracy on the other and present numerical simulation and optimization results based on our model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High frequency and numerical Eulerian methods for aeroacoustic problems

The aim of this work is the simulation of the acoustic propagation in a moving flow using the high-frequency approach. We linearize the Euler equations around a stationary state for which the resulting system of PDE cannot be in general reduced to a wave equation. We are however able to perform a high-frequency analysis of the acoustic perturbation, using the W.K.B. method, introducing a phase φ$\phi$ and an amplitude A  . The phase φ$\phi$ is solution of a Hamilton–Jacobi equation that we solve by a numerical Eulerian method using a monotone scheme [S.J. Osher, C.W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal, 28(4) (1991) 907–922] following Benamou et al. [A geometric optics method for high frequency electromagnetic fields computations near fold caustics Part I, J. Comput. Appl. Math. 156 (2003) 93–125]. Adopting the techniques of Lax and Rauch [Lectures on Geometric Optics, 〈$〈$http://www.lsa.umich.edu/rauch〉$〉$] for hyperbolic systems, we compute the leading order term of the amplitude A. Our results are still valid in the neighborhood of a fold caustic.     

Unified solutions to some classical problems in rarefied gas dynamics based on the one-dimensional linearized S-model equations

An analytical version of the discrete-ordinates method is used to derive solutions for a class of problems defined in terms of the S-model kinetic equations. In addition to a general derivation, which is common to all the problems, specific analytical and computational aspects for each one of the problems are presented. In particular, numerical results for velocity profile, heat-flow profile and flow rates are obtained with high accuracy for the plane channel problems. In the case of half-space problems, the thermal and viscous-slip coefficients are also listed. Received: September 20, 2004; revised: March 31, 2005     

Numerical verification of delta shock waves for pressureless gas dynamics

The subject of this paper is theoretical analysis and numerical verification of delta shock wave existence for pressureless gas dynamic system. The existence of overcompressive delta shock wave solution in the framework of Colombeau generalized functions is proved. This result is verified numerically by specially designed procedure that is based on wave propagation method implemented in CLAWPACK. The method is coupled with dynamic refinement mesh. We also consider a strictly hyperbolic system obtained from the original one by perturbation and change of variables. The same numerical procedure is applied to the perturbed problem. The obtained numerical results in both cases confirm theoretical expectations.     

Numerical analysis of oscillatory Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for a hard sphere molecular gas

The time-dependent motion of a rarefied gas between two parallel planes caused by an oscillatory motion of the plane is studied based on the linearized Boltzmann equation for a hard sphere molecular gas. With the aid of a deterministic numerical method, an accurate numerical analysis is carried out for a wide range of gas rarfaction and oscillatory frequency. The detailed data of the shear stress acting on the planes is provided in a complete form for a wide range of the parameters. The transition of the solution from low to high frequencies under various degrees of gas rarfaction is discussed.     

Numerical analysis of oscillatory Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for a hard sphere molecular gas

The time-dependent motion of a rarefied gas between two parallel planes caused by an oscillatory motion of the plane is studied based on the linearized Boltzmann equation for a hard sphere molecular gas. With the aid of a deterministic numerical method, an accurate numerical analysis is carried out for a wide range of gas rarfaction and oscillatory frequency. The detailed data of the shear stress acting on the planes is provided in a complete form for a wide range of the parameters. The transition of the solution from low to high frequencies under various degrees of gas rarfaction is discussed.     

Modified iteration method for solving fractional gas dynamics equation

This paper is devoted to the time‐fractional gas dynamics equation with Caputo derivative. Fractional operators are very natural tools to model memory‐dependent phenomena. Modified iteration method is proposed to obtain the approximate and analytical solution of the fractional gas dynamics equation. This method is a combined form of the new iteration method and Laplace transform. Modified iteration method really is powerful and simple method compared with other methods. Existence and uniqueness of solution are proven. Numerical results for different cases of the equation are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem

In this paper, based on a two-grid method and a recent local and parallel finite element method, a parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem is proposed and analyzed. This method ensures that all the local subproblems on the fine grid can be solved in parallel. Optimal error bounds of the approximate solution are obtained. Finally, numerical experiments are presented to demonstrate the accuracy and effectiveness of the proposed method.     

Numerical solutions for jump-diffusions with regime switching

This paper is devoted to numerical solutions for a class of jump-diffusions with regime switching. After briefly reviewing the notion of jump-diffusions with regime switching, finite-difference procedures are constructed. Under simple conditions, it is proved that the algorithm converges to the desired limit by means of a martingale problem formulation. Numerical experiments are carried out to demonstrate the performance of the algorithm.     

Numerical solution of navier-stokes equations for viscous gas by the semimarching method

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 225–236.     

Numerical differentiation for high orders by an integration method

This paper mainly studies the numerical differentiation by integration method proposed first by Lanczos. New schemes of the Lanczos derivatives are put forward for reconstructing numerical derivatives for high orders from noise data. The convergence rate of these proposed methods is as the noise level δ→0. Numerical examples show that the proposed methods are stable and efficient.     

Numerical scheme for multilayer shallow-water model in the low-Froude number regime

The aim of this note is to present a multi-dimensional numerical scheme approximating the solutions to the multilayer shallow-water model in the low-Froude-number regime. The proposed strategy is based on a regularized model where the advection velocity is modified with a pressure gradient in both mass and momentum equations. The numerical solution satisfies the dissipation of energy, which acts for mathematical entropy, and the main physical properties required for simulations within oceanic flows.     

A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

The boundary integral method for the linearized rotating Navier-Stokes equations in exterior domain

In this paper, we apply the boundary integral method to the linearized rotating Navier-Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the linearized rotating Navier-Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence and uniqueness of solution. Finally, we study the finite element approximation for the coupled problem and obtain the error estimate between the solution of the coupled problem and its approximation solution.