【作者】 李涛；

【导师】 姜同松； 傅希林；

【作者基本信息】 山东师范大学 ， 应用数学， 2012， 硕士

【摘要】 薛定谔方程(Schronger Equation)是由奥地利物理学家薛定谔提出的量子力学中的一个基本方程,是将物质波的概念和波动方程相结合建立的偏微分方程,它揭示了微观物理世界物质运动的基本规律,在原子,分子,固体物理,核物理,化学等领域中被广泛应用.因此,讨论这类方程的数值解法,具有重要的理论和现实意义.对于物理现象的模拟,无网格方法是相对较新的数值算法.近些年来,无网格方法已经被广泛的应用到流体力学,机械工程等领域.在本文中,主要讨论了两种无网格方法,Kansa’s方法与特解方法(MPS),并将其应用到薛定谔方程的数值解法.以上两种数值算法的有效性在本文中得到了良好的体现.过去的几十年中,传统的数值算法,例如有限差分方法[34],有限元方法[6].边界元方法[40]等,在工程和科学分支中取得了巨大的成功.这种基于网格的方法再将来依然会起到关键的作用,但是在求解的过程中,它们必须要在求解区域上产生一个网格.相对而言,无网格方法可以直接在区域上进行求解,从而避免了刻画网格带来的困难.更多还原

【Abstract】 Schro¨nger equation is a basic function in the quantum mechanics which is proposedby Schro¨nger, and it is a kind of partial diferential equation(PDE) combining conceptionof material wave with wave equation. Schro¨nger equation reveals basic law of materialmotion in the micro-physics world, and is extensively applied in the atom,fraction,solidphysics,nuclear physics,chemistry and other fields.Therefore,the discussion of the equa-tion has important theoretical and practical significance.Meshless methods are relatively new numerical algorithms for the simulation ofphysical phenomena.In recent years,meshless methods have been extensively applied toproblems in fluid dynamics,solid mechanics, and other fields.In this dissertation, two ver-sions of meshless methods, Kansa’s Method and Particular Solution Method(MPS), havebeen developed and applied to solving Schro¨nger equation.And the valid of methods areproved in this work.Traditional simulation algorithms, such as the Finite Diference Method(FDM),theFinite Element Method(FEM) and the Boundary Element Method(BEM),have achievedgreat success in many branches of engineering and the sciences during the past severalyears.These mesh based methods will continue to play central roles in the future,but theyall require that a mesh or grid be generated for the domain as part of the solution pro-cess.Meshless method, in contrast,use the geometry of the domain directly to avoid manyof difculties.更多还原