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

本提案的主要重点是发展和分析一种适用于双曲守恒律的高阶数值格式的新的参数化最大原理保持通量限制器技术。我们还将设计一种通量限制技术来获得高阶保正格式。保持最大原理和正性的数值格式是可取的，因为物理上相关的解具有这些性质。该方法的发展基于有限差分方法，具有计算成本低，计算精度高的优点，特别是在多维模拟中。在提出的框架内，可以设计出保留高阶有限差分、有限体积和不连续Galerkin格式的保守极大值原理，从而允许显着大的CFL数，从而提高计算模拟效率。本文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，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的缺点，并在许多关键应用中允许更有效、更健壮和更准确的计算机模拟。其中一个应用是超音速流动问题，它在设计天体物理射流和再入飞行器的仿真中具有重要意义。另一个应用是磁流体动力系统的研究，它出现在空间天气建模、电力推进源建模和涉及等离子体的系统（如等离子体打开开关、通过等离子体的飞行控制、等离子体辅助燃烧）中。这些问题对于设计具有重大工业和商业价值的下一代器件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.