Efficient high order methods for two multiscale problems

多尺度问题在工程和物理学中普遍存在。这类问题涉及到发生在各种时间和长度尺度上的现象，这些现象可能在数量级上有所不同。传统的近似方法需要非常精细的网格来解决所有尺度的问题，这对内存和计算时间有很大的要求，从而限制了应用。在本项目中，我们构建了新的多尺度方法来高效、准确地求解两个模型方程，这两个模型方程广泛用于涉及反应的爆炸、燃烧和湍流以及纳米级半导体器件的研究。提出的研究将开发新的多尺度方法，以满足多尺度问题对计算资源日益增长的需求。可靠、高效的多尺度方法将进一步有助于在实际应用中预测物理现象。具体来说，本项目侧重于反应流动方程和薛定谔方程的多尺度方法。在具有多组分和多反应的高速反应流中，在欠分辨网格区域可能出现不正确的不连续传播。我们的方法是将高阶激波捕获方案（如用于对流部分的WENO）与用于反应部分的Harten亚单元分辨率相结合。亚单元处理利用了流动信息，能够控制激波捕获方案的耗散，避免了由于网格不充分分解而产生的伪解。目标是用粗网格在时间和空间上捕捉高速反应流中冲击和不连续的正确位置。在用薛定谔-泊松系统模拟电子输运时，由于溶液的高频振荡，计算量很大。其思想是将解的一些已知结构合并到不连续伽辽金方法的基函数中。这可以通过建立基于半经典近似WKB渐近的局部解空间来实现，该解空间具有一定的解的多尺度结构。我们的目标是构建一个廉价可靠的薛定谔-泊松系统求解器来模拟纳米级半导体中电子的量子输运。

Multiscale problems are ubiquitous in engineering and physics. This kind of problem involves phenomena that occur across a variety of time and length scales, which may vary in orders of magnitude. To prevent inaccurate solutions, traditional approximation methods need extremely refined meshes to resolve all the scales, which places huge demands on memory and computation time and thus limits the applications. In this project, we construct new multiscale methods to efficiently and accurately solve two model equations that are broadly used in studies of detonation, combustion and turbulence involving reactions, and nanoscale semiconductor devices. The proposed research will develop new multiscale methods to meet with the increasing demand for computational resources in multiscale problems. Reliable and efficient multiscale methods will further help predict physical phenomena in realistic applications. Specifically, this project focuses on multiscale methods for reactive flow equations and the Schrodinger equation. In high-speed reacting flows with multispecies and multireactions, incorrect propagation of discontinuities may occur in underresolved mesh regions. Our approach is to combine a high order shock-capturing scheme such as WENO for the convection part with Harten's subcell resolution for the reaction part. The subcell treatment utilizes the flow information and is able to control the dissipation of shock-capturing schemes to avoid the spurious solutions due to the underresolved mesh. The goal is to capture the correct locations of shocks and discontinuities in high-speed reacting flows with coarse meshes in both time and space. In simulations of electron transport modeled by the Schrodinger-Poisson system, the computational cost is huge due to the high frequency oscillations of the solution. The idea is to incorporate some known structures of the solution into the base functions of Discontinuous Galerkin methods. This can be accomplished by building local solution spaces based on the semiclassical approximation WKB asymptotic, which has certain multiscale structures of the solution. We aim to construct an inexpensive and reliable solver for Schrodinger-Poisson system to simulate quantum transport of electrons in nanoscale semiconductors.