Efficient high order methods for two multiscale problems

多尺度问题在工程和物理中普遍存在。这类问题涉及在各种时间和长度尺度上发生的现象，这些现象可能在数量级上有所不同。为了防止不准确的解决方案，传统的近似方法需要非常精细的网格来解决所有的尺度，这对内存和计算时间提出了巨大的要求，从而限制了应用。在这个项目中，我们构建了新的多尺度方法来有效和准确地求解两个模型方程，这两个模型方程被广泛用于爆炸，燃烧和湍流反应以及纳米半导体器件的研究。该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。可靠和有效的多尺度方法将进一步帮助预测实际应用中的物理现象。具体来说，这个项目的重点是反应流方程和薛定谔方程的多尺度方法。在多组分多反应的高速反应流中，不连续性的不正确传播可能发生在欠分辨网格区域。我们采用联合收割机将对流部分的高阶激波捕捉格式韦诺与反应部分的Harten子胞方法相结合。亚胞处理利用了流动信息，并能够控制激波捕捉方案的耗散，以避免由于欠分辨网格而导致的伪解。其目标是捕捉正确的位置的冲击和不连续的高速反应流与粗网格在时间和空间。在用Schrodinger-Poisson系统模拟电子输运问题时，由于解的高频振荡，计算量很大。我们的想法是将一些已知的结构的解决方案的不连续Galerkin方法的基函数。这可以通过基于半经典近似WKB渐近建立局部解空间来实现，该近似具有一定的解的多尺度结构。我们的目标是构建一个廉价和可靠的薛定谔-泊松系统的求解器来模拟纳米半导体中的电子量子输运。

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.