CAREER: Towards Robust and Efficient High-Order Adaptive Computational Methods for Conservation Laws in Complex Geometries -- Analysis, Implementation, and Applications

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The purpose of the CAREER effort is to take some important steps towards the development and application of new, adaptive, high-order accurate methods for solving conservation laws. While the technical emphasis will be on efficient implicit time-stepping methods and techniques for local error and regularity estimation, the effort will maintain a focus on large scale, realistic applications from the applied sciences and engineering, and the use of national high performance computing facilities. The complexity of this multidisciplinary task, the development of a computational platform focusing on educational elements, and the fundamental nature of the problems being considered will provide a stimulating environment in which to train future computational scientists in emerging computational techniques for solving conservation laws.Application of basic physical principles of conservation of mass, momentum, and energy has proven itself to lead to accurate and reliable mathematical models of the physical world surrounding us. Such models, known as conservation laws, display a richness, much like the physical world, that continues to challenge developers of computational techniques for solving such problems. The mere universality of such models, however, warrants that new and improved methods be developed to enable the efficient and robust modeling of problems in areas as diverse as the dynamics of fluids and gases, optical communication and high-speed electronics, applications in electromagnetic radiation, or even climate modeling. Hesthaven proposes to develop new adaptive high-order methods for such problems and to train computational scientists in this area.