Mathematical Sciences: Nonlinear Waves: Theory and Computation

非线性现象在所有科学中都很重要。 它们构成了一个广泛而知之甚少的领域， 各种各样的数学学科。 本研究 程序地址到非线性分析的一个区域， 最近显示出相当大的进展：非线性波在 双曲系统和耦合双曲椭圆系统。 研究人员打算开发计算方法， 都是基于这个数学理论， 这些波的数学理论的发展。 这项工作的目标是发展数学理论 并在最大限度内加以利用。 计算过程 前沿跟踪的成功表明， 这个目标是可以实现的。 其中一个调查员的专业 这些成就是科学原理的证明， 前方跟踪法 此方法现在已被接受， 在研究计划中开发的证据基础。 的 独立工作证实了前沿跟踪的成功 他人 它在两个维度上与之前的成功相似， 这是通过随机选择方法在一个 维度 数学理论的发展和应用 计算过程同样是一个典型的特征， 随机选择法

Nonlinear phenomena are important across all of science. They constitute a broad and poorly understood area which draws on a diverse range of mathematical disciplines. This research program addresses to an area of nonlinear analysis which has recently shown considerable progress: nonlinear waves in hyperbolic systems and in coupled hyperbolic-elliptic systems. The investigators intend to develop computational methods which are based on this mathematical theory and to participate in the development of the mathematical theory of these waves. The goal of this work is to develop the mathematical theory and to utilize it to the maximum extent possible within the computational process. The success of front tracking shows that this goal is attainable. One of the investigator's major accomplishments has been the proof of scientific principle for the front tracking method. This method is now accepted, on the basis of evidence developed within the research program. The success of front tracking has been confirmed by independent work of others. It parallels in two dimensions the previous success which has been achieved by the Random Choice Method in one dimension. The development and use of mathematical theory in the computational process was likewise a characteristic feature of the Random Choice Method.