HIGH-ORDER INVARIANT DOMAIN PRESERVING NUMERICAL METHODS FOR NONLINEAR HYPERBOLIC SYSTEMS

许多重要的物理现象都是用双曲守恒定律的非线性系统来模拟的。当逼近这些问题时，人们会遇到尖锐的界面、接触不连续面、冲击和其他非线性波相互作用。在这些区域，高阶近似方法不稳定，并表现出振荡。控制这些振荡是非常重要的，因为在数值模拟中保持解的质量、正性或有界性是至关重要的。此外，已知虚假振荡会促进收敛到非物理弱解或简单地导致无法产生近似。在这个方向上的任何进步都将产生广泛的影响，因为我们想要解决的问题涉及工程（机械、航空航天、核、海洋等）、环境科学、地球物理、石油工程等许多领域。提出一种新的鲁棒近似技术来解决发展冲击或尖锐界面的非线性双曲问题，将对控制或处理这类现象仍然是一个巨大挑战的科学和工程的各个领域产生影响。本研究的重点是研究和开发新的高阶逼近技术，用于任何空间维度的非结构化网格上的非线性双曲系统。该项目将围绕三个主要目标进行组织：(1)发展标量守恒方程的高阶最大原理保持方法。重点将放在至少在空间和时间上的三阶数值方法的设计和分析上。将建立其中一些方法的收敛估计；(2)设计了非均匀网格上任意空间维的任意双曲系统的不变保域方法。目标是构建至少在空间和时间上具有二阶精度的不变域保留方法。目标是要有方法，以保持所有的不变域的基础物理系统；(3)该项目的最后一部分将包括将新方法扩展到具有源项的系统，如浅水方程和辐射输运。

Many important physical phenomena are modeled by nonlinear systems of hyperbolic conservation laws. When approximating such problems one encounters sharp interfaces, contact discontinuities, shocks and other nonlinear wave interactions. In such regions high order approximation methods are not stable and exhibit oscillations. It is very important that these oscillations be controlled because preservation of mass, positivity or boundedness of the solution is critical in numerical simulations. Moreover, spurious oscillations are known to promote convergence to nonphysical weak solutions or simply lead to failure to produce an approximation. Any advance in this direction will have a broad impact insofar the class of problems we want to address touches many fields in engineering (mechanical, aerospace, nuclear, ocean, etc.), in environmental sciences, in geophysics, in petroleum engineering, etc. Proposing a novel robust approximation technique for solving nonlinear hyperbolic problems developing shocks or sharp interfaces will have impact in every areas of science and engineering where controlling or dealing with this type of phenomenon is still an enormous challenge.The main focus of this proposal to investigate and develop new high-order approximation techniques for nonlinear hyperbolic systems on unstructured meshes in any space dimension. The project will be organized around three main objectives: (1) Development of high-order maximum principle preserving methods for scalar conservation equations. The emphasis will be on the design and analysis of numerical methods that are at least third-order in space and time. Convergence estimates for some of these methods will be established; (2) Design of invariant domain preserving methods for any hyperbolic system in any space dimension on non-uniform meshes. The goal is to construct invariant domain preserving methods that are at least formally second-order accurate both in space and time. The objective is to have methods that preserve all the invariant domains of the underlying physical system; (3) The last part of the project will consists of extending the new methodology to systems with source terms like the shallow water equations and radiative transport.