Development, Analysis and Application of Numerical Methods for Nonlinear Partial Differential Equations

9706827 Stanley Osher 本研究系关于偏微分方程之“不规则”解之精确且有效率之计算。 这包括具有不连续性、奇异性、精细尺度结构或持续振荡的解。 主要议题包括：（1）非线性偏微分方程的动力学公式及其应用;（2）通过水平集方法捕获界面;（3）基于守恒律数值解思想的不连续性捕获;（4）振荡和临界阈值现象的数值和分析研究。 拟议的研究将影响许多科学和技术领域。 随着现代计算机的出现，以前难以解决的问题可以得到精确的解决。 当然，这需要精确和收敛的算法，这些困难的非线性和计算密集的问题。 该小组以前开发的算法已经在全国各地的国家实验室和工业中广泛使用。 这里提出的方法将是有用的，在主机的应用，包括燃烧，石油回收，晶体生长，电磁和声散射，薄膜半导体生长，飞机设计，仅举几例。

9706827 Stanley Osher This research is concerned with the accurate and efficient computation of "irregular" solutions of partial differential equations (PDEs). This includes solutions with discontinuities, singularities, fine scale structure, or persistent oscillations. The main topics include (1) kinetic formulations of nonlinear PDE's and applications; (2) interface capturing through the level set method; (3) discontinuity capturing based on ideas developed for the numerical solution of conservation laws; and (4) numerical and analytical study of oscillations and critical threshold phenomena. The proposed research will impact numerous areas of science and technology. With the advent of modern computers, formerly intractable problems can be solved accurately. This, of course, requires accurate and convergent algorithms for these difficult nonlinear and computationally intense problems. The algorithms formerly developed by this group are already in wide use throughout the country in national laboratories and industry. The proposed methods to be developed here will be useful in a host of applications including combustion, oil recovery, crystal growth, electromagnetic and acoustic scattering, thin film semiconductor growth, aircraft design, to name just a few.