Mathematical Sciences: Nonlinear Waves: Theory and Computation

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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.