High Order Maximum Principle Preserving Finite Difference Schemes for Hyperbolic Conservation Laws

该建议的主要重点是开发和分析一种新的参数化最大值原理保持通量限制器技术的高阶数值格式应用于双曲守恒律。一个通量限制技术也将被设计来获得高阶正性保持格式。保持最大值原理和正性的数值方案是可取的，因为物理相关的解决方案具有这些属性。开发是基于有限差分方法，它具有产生精确的近似，计算成本低，特别是在多维模拟的优势。在所提出的框架内，保守的最大值原理保持高阶有限差分，有限体积和间断Galerkin格式可以设计，允许显着大CFL数，因此更有效的计算模拟。该计划研究的一些重要应用包括可压缩欧拉方程、磁流体动力学方程和Vlasov-Maxwell方程，研究人员正在开发新的计算技术，这些技术可以应用于科学和工程中的困难和非常重要的问题。这些技术解决了现有方法中的缺点，并应允许在许多关键应用中进行更有效，更鲁棒和更准确的计算机模拟。超声速流动问题就是其中的一个应用，它在设计天体物理喷流和模拟再入飞行器的空间飞行建模中具有重要意义。另一个应用是磁流体动力学系统的研究，它出现在空间天气建模，建模电推进源，并在涉及等离子体的系统（如等离子体开放开关，通过等离子体飞行控制，等离子体辅助燃烧）。这些问题对于具有巨大工业和商业价值的下一代设备的设计具有重要的战略意义。

The main focus of this proposal is to develop and to analyze a novel parametrized maximum principle preserving flux limiter technique for high order numerical schemes applied to hyperbolic conservation laws. A flux limiting technique will also be designed to obtain high order positivity preserving schemes. Numerical schemes that preserve the maximum principle and positivity are desirable because physically relevant solutions have those properties. The development is based on finite difference methods, which have the advantage of producing accurate approximations with low computational cost especially in multi-dimensional simulations. Within the proposed framework, conservative maximum principle preserving high order finite difference, finite volume and discontinuous Galerkin schemes can be designed that allow for significantly large CFL number, and therefore more efficient computational simulation. Some important applications investigated in this proposal include compressible Euler equations, magneto hydrodynamics equations and Vlasov-Maxwell equations.The investigator is developing new computational techniques that can be applied to difficult and very important problems in science and engineering. These techniques address shortcomings in existing methods and should allow more efficient, robust, and accurate computer simulations in a number of critical applications. One such application is the supersonic flow problem, which is of great importance in designing astrophysical jets and also in the simulation of reentry vehicle for space flight modeling. Another application is the study of the magneto hydrodynamic systems, which arises in space weather modeling, in modeling electric propulsion sources, and in systems involving plasma (such as plasma-opening switches, flight control via plasma, plasma assisted combustion). These problems are strategically important for the design of the next generation devices of great industrial and commercial value.