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

许多重要的物理现象都是由双曲型守恒律的非线性系统来模拟的。当近似这类问题时，会遇到尖锐的界面、接触不连续性、冲击和其他非线性波相互作用。在这些区域，高阶近似方法是不稳定的，并且表现出振荡。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，众所周知，虚假振荡会促进收敛到非物理弱解，或者只是导致无法产生近似。在这一方向上的任何进展都将产生广泛的影响，因为我们要解决的问题涉及到工程(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。提出一种新的稳健逼近技术来解决发展激波或尖锐界面的非线性双曲问题，将在科学和工程的各个领域产生影响，在这些领域，控制或处理这类现象仍然是一个巨大的挑战。本建议的主要重点是研究和开发新的高阶逼近技术的非线性双曲系统在任何空间的非结构网格上。该项目将围绕三个主要目标组织：(1)发展标量守恒方程的高阶最大值原理保持方法。重点将是设计和分析至少在空间和时间上是三阶的数值方法。给出了其中一些方法的收敛估计；(2)在非均匀网格上设计了任意空间维上任意双曲型方程组的不变区域保持方法。其目标是构造至少形式上在空间和时间上都是二阶精度的不变区域保持方法。(3)项目的最后部分将包括将新的方法扩展到具有浅水方程和辐射传输等源项的系统。

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.