守恒律的动力学方法及其在计算流体中应用

近年来利用分子运动论研究守恒律方程组的数值方法引起人们广泛关注，它在论证流体力学方程组解的存在性和复杂流体问题的数值模拟等方面起着关键作用。本项目研究内容：多维守恒律方程组的高分辨动力学方法的非线性稳定性、熵条件和收敛性分析；高分辨动力学格式在计算流体力学中的应用；探寻用于低高空流场数值分析的统一算法或混合算法。.

Up to the begin of the nineties of the twentieth century, it becomes very mature to design high-order accurate non-oscillatory algorithms for hyperbolic conservation laws. The researchers have done much more excellent works in this field, and used them to compute numerically large scale scientific problems in engineering. They have become very useful tools for the engineering design and analysis of complex fluid flows. However, high-order accurate non-oscillatory algorithms are much more.non-linear. It is still very difficult to analyze them in theory. To some extent, we may transfer the interesting problem to a linear equation (i.e. discrete or continuous kinetic equations) by using gas-kinetic theory, and decrease the difficulty. Thus,based on the gas-kinetic theory, it is much more theoretically valuable and practically meaningful to study high-order accurate non-oscillatory algorithms and its application in computational fluid dynamics, whose numerical solutions will converge to the physical solutions. The main contents of this project include: convergence, numerical entropy condition, and nonlinear stability of high resolution kinetic schemes for hyperbolic conservation laws; construction of high resolution kinetic schemes for the general governing equations in fluid mechanics and analysis of its mathematical properties and behaviours; applications of high resolution kinetic schemes in numerical.computations of complex physical problems in fluid flows. We have studied numerical entropy condtions, nonlinear stability and.convergence of the high resolution discrete-velocity kinetic schemes－the relaxing schemes for nonlinear hyperbolic conservation laws. The results show that the conservative variable is TVD (total variation diminishing). The solutions of the second order accurate relaxed schemes satisfy a cell entropy inequality for.arbitrary entropy pair. Some remarks on convergence rate are also given. Based.on the pseudoparticle representation of the conservative vectors and fluxes, we.have given positivity-preserving analysis of a class of explicit and implicit.flux-vector splitting schemes. Based on the gas-kinetic theory, we constructed high resolution gas-kinetic schemes for multi-dimensional radiation hydrodynamical.equations, ideal magnetohydrodynamics, and multi-fluid flows, and used successfully them to simulate numericallythree-dimensional flows past space shuttle and resonant oscillations in a tube, and compute complex fluid flow.problems, such as Kelvin—Helmholtz instability etc. The theoretic results obtained via the particle movement may be considered as a reference for the theoretic analysis of generally nonlinear high-order accurate algorithms. The further study of high resolution gas-kinetic schemes is very useful to.study in future a hybrid algorithm that combines DSMC methods with Navier-Stokes.solver, because the high resolution gas-kinetic schemes are simple, robust, and parallelizable, and the direct simulation Monte Carlo (DSMC) type boundary conditions can be implemented naturally according to molecular speeds splitting. As we knew, the study of a hybrid or unified algorithm for fluid flows in continuous, transient, and rarefied flow regions will influence directly the development of aeronautical and astronautical industy.