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

计算建模在科学和工程中用于模拟物理和生物系统的工作方式，以便我们能够更好地理解它们以及如何对它们进行修改以获得社会效益。许多这样的系统混合了平流（或输运）、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，但其中只有少数可以同时处理多个过程。该项目涉及非线性平流-扩散-反应方程系统模拟的另一类数值技术的理论和算法发展。这些新的数值技术显示出巨大的前景，它们可能导致更好的精度和计算效率。对能源生产和环境保护重要的地球科学问题的应用将继续进行。评估、设计和监测涉及地下水库和含水层的人类活动需要对长时间内的平流、扩散和反应过程进行大规模模拟。在能源生产和环境保护方面具有潜在的社会效益。该项目还可能对使用非线性、耦合平流-扩散-反应方程模型的科学和工程的广泛领域产生影响。该项目预计将通过教育和培训两名博士研究生（均为女性，其中一名是当地公民），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有很高的需求。由非线性平流-扩散-反应偏微分方程控制的物理或生物模型的数学结构通常很难理解，而且解可能产生激波或非常陡峭的锋面。该项目涉及用于此类方程系统数值逼近的高阶，隐式，加权基本非振荡（iWENO）格式的理论和算法发展，因为这种类型的格式有可能很好地处理所有三个过程。发展将包括有限体积和有限差分格式，欧拉-拉格朗日方法，和多矩变体。目标是：(1)为问题开发合适的平滑度指标和时间积分器；(2)制定处理可能退化的扩散过程的一般程序；(3)在空间离散化及边界条件处理、满足局部极大值原则等相关问题上取得进展；(4)测试该方法在多孔介质中的应用；(5)在跨学科的环境中教育和训练学生。该项目将导致一个非常通用的计算框架，可以以局部质量保守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的一般计算网格，在空间和时间上都是高阶精度，保持局部质量守恒特性，具有鲁棒性（即无条件线性稳定），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是有效的，因为可以放入缓存存储器的局部信息将主导计算，而离散方程的全局系统将具有尽可能少的自由度。预计该项目将通过该方案的应用对STEM劳动力及其多样性以及地球科学产生更广泛的影响，并可能影响科学和工程的广泛领域，特别是那些使用复杂耦合问题模型的领域，这些模型的应用程序的数学结构可能无法很好地理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.