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

职业生涯的努力的目的是采取一些重要的步骤，发展和应用新的，自适应的，高阶精度的方法来解决守恒定律。虽然技术重点将放在有效的隐式时间步进方法和技术，用于局部误差和规律性估计，但工作重点将继续放在应用科学和工程的大规模实际应用以及国家高性能计算设施的使用上。这一多学科任务的复杂性、侧重于教育要素的计算平台的开发以及所考虑问题的基本性质将提供一个激励性的环境，在此环境中培养未来的计算科学家，使其掌握解决守恒定律的新兴计算技术。和能量已经证明了自己，导致我们周围的物理世界的准确和可靠的数学模型。这些模型，被称为守恒定律，显示了丰富性，就像物理世界一样，继续挑战计算技术的开发人员来解决这些问题。然而，仅仅是这些模型的普遍性，就需要开发新的和改进的方法，以便在流体和气体的动力学、光通信和高速电子学、电磁辐射的应用甚至气候建模等不同领域中对问题进行有效和鲁棒的建模。Hesthaven建议为这些问题开发新的自适应高阶方法，并在这一领域培养计算科学家。

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.