Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations

An approach based on a modified splitting method is proposed for solving the radiation gas dynamics equations in the multigroup kinetic approximation. The idea of the approach is that the original system of equations is split using the thermal radiation transfer equation rather than the energy equation. As a result, analytical methods can be used to solve integrodifferential equations and problems can be computed in the multigroup kinetic approximation without iteration with respect to the collision integral or matrix inversion. Moreover, the approach can naturally be extended to multidimensional problems. A high-order accurate difference scheme is constructed using an approximate Godunov solver for the Riemann problem in two-temperature gas dynamics.     

On Some Finite Difference Scheme for Gas Dynamics Equations

A conservative difference scheme with linear dependence of the pressure on the density of gas is proposed for gas dynamics equations. The scheme allows us to simulate 1-D flows inside a cylindrical domain with time-variable cross-sections and guarantees the positive sign of the density function.     

Analytical solutions to detect the scheme dispersion for the coupled nonlinear equations

It is shown, how even particular traveling wave asymptotic solution may describe the defects on the shock wave profile caused by the dispersion features of the numerical scheme of the coupled nonlinear gas dynamics equations. For this purpose the coupled nonlinear partial differential equations or the so-called differential approximation of the scheme, are obtained, and a simplification of the method of differential approximation is suggested to obtain the desired asymptotic solution. The solution is used to study the roles of artificial viscosity and the refinement of the mesh for the suppression of the dispersion of the scheme.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

On L2-Dissipativity of a Linearized Explicit Finite-Difference Scheme with Quasi-Gasdynamic Regularization for the Barotropic Gas Dynamics System of Equations

Doklady Mathematics - We study an explicit two-level symmetric (in space) finite-difference scheme for the multidimensional barotropic gas dynamics system of equations with quasi-gasdynamic...     

Singularities of multivalued solutions of nonlinear differential equations,and nonlinear phenomena

The purpose of this paper is to use the geometrical theory of nonlinear partial differential equations and the theory of singularities of maps in order to obtain the general scheme for constructing shock waves from multivalued solutions, given by smooth integral manifolds. This scheme is illustrated by some examples from gas dynamics, mechanics, acoustics and thermodynamics.     

The modeling and numerical simulation of gas flow networks

Summary. A network formulation is introduced for the modeling and numerical simulation of complex gas transmission systems like a multi-cylinder internal combustion engine. Several simulation levels are discussed which result in different network representations of a specific system. Basic elements of a network are chambers of finite volume, straight pipes and connections like valves or nozzles. The pipe flow is modeled by the unsteady, one-dimensional Euler equations of gas dynamics. Semi-empirical approaches for the chambers and the connections yield differential-algebraic equations (DAEs) in time. The numerical solution is based on a TVD scheme for the pipe equations and a predictor-corrector method for the DAE-system. Simulation results for an internal combustion engine demonstrate the practical interest of the new approach. Received May 12, 1994 / Revised version received August 26, 1994     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

On the modeling and simulation of boundary flow through partially open pipe ends

The determination of boundary conditions for the Euler equations of gas dynamics in a pipe with partially open pipe ends is considered. The boundary problem is formulated in terms of the exact solution of the Riemann problem and of the St. Venant equation for quasi-steady flow so that a pressure-driven calculation of boundary conditions is defined. The resulting set of equations is solved by a Newton scheme. The proposed algorithm is able to solve for all inflow and outflow situations including choked and supersonic flow.Received: August 7, 2002; revised: November 11, 2002     

Une méthode numérique décentrée pour la résolution d'écoulements diphasiques

This Note focuses on the numerical approximation of two-fluid flow models described by six balance equations. We introduce an original splitting technique especially derived to use the approximate Riemann solvers of the usual gas dynamics and to allow for a straightforward extension to various and detailed exchange source terms. When based on suitable kinetic upwind schemes, the whole scheme preserves the positivity of all the thermodynamic variables under a fairly unrestrictive “CFL like” condition. Several stiff numerical teats, are presented including phase separation, in order to highlight the efficiency of the proposed method.     

Implicit-explicit schemes for flow equations with stiff source terms

In this paper, we design stable and accurate numerical schemes for conservation laws with stiff source terms. A prime example and the main motivation for our study is the reactive Euler equations of gas dynamics. Furthermore, we consider widely studied scalar model equations. We device one-step IMEX (implicit-explicit) schemes for these equations that treats the convection terms explicitly and the source terms implicitly.For the non-linear scalar equation, we use a novel choice of initial data for the resulting Newton solver and obtain correct propagation speeds, even in the difficult case of rarefaction initial data. For the reactive Euler equations, we choose the numerical diffusion suitably in order to obtain correct wave speeds on under-resolved meshes.We prove that our implicit-explicit scheme converges in the scalar case and present a large number of numerical experiments to validate our scheme in both the scalar case as well as the case of reactive Euler equations.Furthermore, we discuss fundamental differences between the reactive Euler equations and the scalar model equation that must be accounted for when designing a scheme.     

Entropy solutions of the compressible Euler equations

We consider the three-dimensional Euler equations of gas dynamics on a bounded periodic domain and a bounded time interval. We prove that Lax–Friedrichs scheme can be used to produce a sequence of solutions with ever finer resolution for any appropriately bounded (but not necessarily small) initial data. Furthermore, with some technical assumptions, e.g. that the density remains strictly positive in the sequence of solutions at hand, a subsequence converges to an entropy solution. We provide numerical evidence for these results by computing a sensitive Kelvin–Helmholtz problem.     

Piecewise parabolic method on local stencil for gasdynamic simulations

A numerical method based on piecewise parabolic difference approximations is proposed for solving hyperbolic systems of equations. The design of its numerical scheme is based on the conservation of Riemann invariants along the characteristic curves of a system of equations, which makes it possible to discard the four-point interpolation procedure used in the standard piecewise parabolic method (PPM) and to use the data from the previous time level in the reconstruction of the solution inside difference cells. As a result, discontinuous solutions can be accurately represented without adding excessive dissipation. A local stencil is also convenient for computations on adaptive meshes. The new method is compared with PPM by solving test problems for the linear advection equation and the inviscid Burgers equation. The efficiency of the methods is compared in terms of errors in various norms. A technique for solving the gas dynamics equations is described and tested for several one-and two-dimensional problems.     

Splitting schemes for the ocean dynamics equations

A splitting scheme in physical processes is proposed for a system of large-scale ocean dynamics equations. The convergence to an exact solution is proved for this scheme.     

Numerical simulation of the unsteady aerodynamic interaction of two plane airfoil cascades

A method for the calculation of unsteady aerodynamic interaction of two plane airfoil cascades that are in relative motion in a subsonic flow of ideal gas is developed. This interaction provides a two-dimensional approximation of the flow in a stage of an axial turbomachine. The method is based on the reduction of the problem to the calculation of the unsteady flow in a single interblade passage of each of the cascades. The calculation uses generalized space-time periodicity relations corresponding to the unsteady process of interest. The calculation is based on the direct numerical integration of the non-stationary gas dynamics equations with the use of the finite difference Godunov-Kolgan-Rodionov scheme of the second approximation order with respect to time and space. The calculation procedure includes the determination of the acoustic fields that are generated by the stage in the incident flow and in the flow behind it. The results of the calculations that illustrate the accuracy of the numerical solution and the capabilities of the method are presented.     

A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Summary. We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of , allows us to pass to a limit as . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme. Received February 7, 2000 / Published online December 19, 2000     

A cell-centered lagrangian scheme in two-dimensional cylindrical geometry

A new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed by Maire et al.The main new feature of the algorithm is that the vertex velocities and the numerical fluxes through the cell interfaces are all evaluated in a coherent manner contrary to standard approaches.In this paper the method introduced by Maire et al.is extended for the equations of Lagrangian gas dynamics in cylindrical symmetry.Two different schemes are proposed,whose difference is that one uses volume weighting and the other area weighting in the discretization of the momentum equation.In the both schemes the conservation of total energy is ensured,and the nodal solver is adopted which has the same formulation as that in Cartesian coordinates.The volume weighting scheme preserves the momentum conservation and the area-weighting scheme preserves spherical symmetry.The numerical examples demonstrate our theoretical considerations and the robustness of the new method.     

Three-and four-step implicit absolutely stable fourth-order Runge-Kutta schemes

Two types of implicit fourth-order Runge-Kutta schemes are constructed for first-order ordinary differential equations, multidimensional transfer equations, and compressible gas equations. The absolute stability of the schemes is proved by applying the principle of frozen coefficients. Adaptive artificial viscosity ensuring good time convergence and oscillations damping near discontinuities is used in solving gas dynamics equations. The comparative efficiency of the schemes is illustrated by numerical results obtained for compressible gas flows.     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.