Implicit Weighted Essentially Non-Oscillatory (WENO) Schemes for Advection-Diffusion-Reaction Systems

计算模型在科学和工程中被用来模拟物理和生物系统如何工作，以便我们能够更好地理解它们以及如何为了社会利益而对它们进行修改。许多这样的系统混合了平流(或输送)、扩散和反应过程。我们有很好的数值技术来模拟单个这样的过程，但其中只有几个可以同时处理多个过程。该项目涉及用于模拟非线性对流-扩散-反应方程组的另一类数值技术的理论和算法的发展。这些新的数值技术显示出巨大的前景，它们可能会导致更高的精度和计算效率。将继续应用于对能源生产和环境保护具有重要意义的地学问题。对涉及地球地下水库和含水层的人类活动的评估、设计和监测需要对长期的平流、扩散和反应过程进行大规模模拟。能源生产和环境保护具有潜在的社会效益。该项目还可能对使用由非线性、耦合的对流-扩散-反应方程组成的模型的科学和工程的广泛领域产生影响。预计该项目将通过教育和培训两名博士研究生(均为女性，一名为土著公民)，对STEM劳动力及其多样性产生影响。这样的学生在工业和政府实验室以及学术界都很受欢迎。由非线性对流-扩散-反应偏微分方程式控制的物理或生物模型的数学结构通常很难理解，解决方案可能会产生冲击或非常陡峭的前沿。本项目涉及高阶隐式加权基本无振荡(IWENO)格式的理论和算法发展，用于此类方程组的数值逼近，因为这种类型的格式有可能很好地处理所有三个过程。其发展将包括有限体积和有限差分格式、欧拉-拉格朗日方法和多矩变量。其目标是：(1)开发适合问题的光滑度指示器和时间积分器；(2)开发处理可能退化的扩散过程的一般程序；(3)在空间离散化和相关问题上取得进展，例如处理边界条件和满足局部极大值原理；(4)测试应用于多孔介质的方法；以及(5)在跨学科环境中教育和培训学生。该项目将导致一个非常通用的计算框架，可以以局部质量守恒的方式近似所有必要的物理。它将很容易实现，处理二维和三维空间中的通用计算网格，在空间和时间上都是高精度的，保持局部质量守恒性质，是健壮的(即，无条件线性稳定的)，并最大限度地提高网格分辨率。这些方案在内存带宽有限的高性能计算机上将是有效的，因为高速缓存中可以容纳的局部信息将主导计算，而全局离散方程系统将具有尽可能少的自由度。该项目预计将对STEM劳动力及其多样性产生更广泛的影响，并通过计划的应用对地球科学产生更广泛的影响，它可能会影响到科学和工程的广泛领域，特别是那些使用复杂、耦合问题的模型的科学和工程领域，对于这些问题，应用程序的数学结构可能不被很好地理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Computational modeling is used in science and engineering to simulate how physical and biological systems work, so that we can better understand them and how they may be modified for societal benefit. Many of these systems mix advective (or transport), diffusive, and reactive processes. We have good numerical techniques for simulating a single such process, but only a few of these can handle multiple processes at once. This project concerns theoretical and algorithmic development of an alternate category of numerical techniques for simulation of systems of nonlinear advection-diffusion-reaction equations. These new numerical techniques show great promise, and they are likely to lead to better accuracy and computational efficiency. Applications to geoscience problems important to energy production and environmental protection will be pursued. Assessment, design, and monitoring of human activities involving reservoirs and aquifers in the Earth's subsurface require large-scale simulation of advective, diffusive, and reactive processes over long time periods. There is a potential societal benefit in energy production and environmental protection. The project may also have an impact on broad areas of science and engineering that use models consisting of nonlinear, coupled advection-diffusion-reaction equations. The project is expected to have an impact on the STEM workforce and its diversity through the education and training of two Ph.D. graduate students (both female, one a native citizen). Such students are in high demand in industrial and governmental labs, as well as in academia.The mathematical structure of physical or biological models governed by nonlinear advection-diffusion-reaction partial differential equations is often poorly understood, and solutions can develop shocks or very steep fronts. This project concerns theoretical and algorithmic development of high order, implicit, weighted essentially non-oscillatory (iWENO) schemes for numerical approximation of systems of such equations, because this type of scheme has the potential to handle all three processes well. The development will including finite volume and finite difference schemes, Eulerian-Lagrangian approaches, and a multi-moment variant. The objectives are to (1) develop a suitable smoothness indicator and time integrator for the problem; (2) develop a general procedure to handle possibly degenerate diffusive processes; (3) make advances on space discretization and related issues, such as handling boundary conditions and satisfying local maximum principles; (4) test the approach on applications to porous media; and (5) educate and train students in an interdisciplinary setting. The project will lead to a very general computational framework can approximate all the necessary physics in a locally mass conservative way. It will be simple to implement, handle general computational meshes in two and three space dimensions, be high order accurate in both space and time, maintain local mass conservation properties, be robust (i.e., unconditionally linearly stable), and maximize mesh resolution. The schemes will be efficient on high performance computers, which are memory bandwidth limited, because local information that can fit in cache memory will dominate the computations, and the global system of discrete equations will have about as small a number of degrees of freedom as possible. The project is expected to have a broader impact on the STEM workforce and its diversity, and on the geosciences through applications of the schemes, and it may impact broad areas of science and engineering, especially those that use models of complex, coupled problems for which the mathematical structure of the application may not be well understood.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.