Hydrodynamic Equations And The Diffusion Equation Finite Point Method

Studies on numerical methods for compressible multi-material radiation (mag-netic) hydrodynamics problem are important topics in inertial confinement fusion, Z-pinch, weapon physics and other research fields. Currently, the grid-based Lagrangian method and ALE method are the main numerical methods for solving these problems. However, large deformation problem of the meshs is always a bottleneck in computing. To develop meshfree method is an important way for shaking off the shackles of the grid. Finite point method based on the directional difference is a kind of meshfree finite difference method. In this dissertation, numerical methods based on finite point method for compressible radiation hydrodynamics were studied. Two of the key issues were addressed. One is to explore Lagrangian finite point method for two-dimensional compressible fluid dynamics problems; the other is searching the corresponding finite point method for diffusion equation. They are two important components of the over-all calculation of radiation hydrodynamics. The main work of this dissertation is as follows.1. According to the solvability of the finite point formula (five-point formula) and the feature of the problem considered, a robust, efficient neighbor points selec-tion strategy was demonstrated. The concept of angular domain was firstly pre-sented. Based on the idea of taking the nearest point in effective angular domain, three kinds of neighbor points selection algorithm were proposed. They are four-directional angular domain method, omni-directional angular domain method and three-directional angular domain method. A suitable data structure was designed. The independent program modules were established. It provides the basic tools for the study of finite point method.2. For one dimensional compressible multifluids, a Lagrangian finite point method was presented. The proposed method is a meshfree numerical procedure based on the combination of interior points scheme and interface point tracking algorithm. The discretization of the unknown function and its derivatives are defined only by the position of the so called Lagrangian points. The interior point formula-tion is based on Taylor series expansion in the continuous regions on both sides of the interface. Unlike most current meshfree method, initially, a point is settled at the interface position. Then the state of interface point is updated using the Rankine-Hugoniot conditions at the interface together with characteristics differ-ence computation. This interface tracking algorithm is the main new features of the method. Numerical tests demonstrate the accuracy of the method.3. A Lagrangian finite point method for two-dimensional compressible hydrodynamics problem was studied. Similar to the one-dimensional approach, initially, The pro-posed method is consisted of interior points scheme and boundary point algorithm. Three kinds of discrete schemes, namely, Godunov-type scheme, center-type scheme and upwind scheme for interior points and boundary points were constructed by using five-point approximation formula of the gradient and the divergence and the appropriate neighbor points selection algorithm, and combining with the designing skills of grid-based difference scheme. Numerical results verify the effectiveness of the algorithms.4. Based on the characteristic theory of hyperbolic systems, by decomposing the ve-locity vector in two non-colinear directions and introducing two new variables, the so called canonical characteristic relations for two dimensional Euler equations were firstly derived. These relations retain the derivatives strictly along the bichar-acteristic directions, and can be viewed as the 2D extension of the characteristic relations in 1D case. These relations are the theoretical basis of the finite point method for two dimensional compressible multi-material flow problems.5. Based on the numerical direction differential formulas and effective neighbor points selection strategy, the finite point schemes based on a five-point formula for the multi-material diffusion problem with large deformation were constructed and the finite point method for Neumann boundary was studied. Discrete maximum prin-ciple was also proved. For multi-material problem, the discrete points were also settled at the interface and the finite point scheme was designed according to the connective conditions of the temperature and heat flux on the interface. The ex-plicit, implicit and precise integration algorithm were used to discrete the equations in time direction. Numerical results verify the validity of the algorithm.6. Scattered data approximation in the plane was studied by using finite point method. Based on the first order numerical directional derivative formula and the directional Taylor expansion formula, various local parameterless finite point approximation formulas with different approximate accuracy were deduced, which can be used for scattered data approximation in the plane.