浅水中污染物模型的保正WENO格式及其快速算法

The existing numerical methods for pollutant models in shallow water can not preserve the water depth and pollutant concentration to be positive simultaneously,which is difficult to handle the dry/wet interface and the very low pollutant concentration,and do not have high oder accuracy in the smooth region，lack high resolution property near the strong discontinuity at the same time.In recent years, a growing high order positivity-preserving numerical methods provide us with a new perspective for the numerical simulation about the pollutant models in shallow water. In order to carry out more efficient numerical simulation,this project deals with the study of pollutant models, based on the applicant's fundamental research on pollutant model and on body-fitted meshes. The applicant propose a improved finite volume WENO schemes based on unstructured meshes,which can maintain the exact balance between the general source term and the advection term, mainmain the water depth and the pollutant concentration to be positive simultaneously, obtain high order accuracy and high resolution property, enjoy higher computational efficiency. The scientific interest of this project is to uncover the pollutant's motion laws in shallow water from a new angle of view, profoundly clarify the complex mechanism of the pollutant transport process, and its results would supplement the theories and methods for the domain of computational fluid dynamics further, meanwihle, provide the water resource conservation with new thoughts and technological assistances, and have potential application prospect in ecological conservation domain.

浅水中污染物模型的现有数值方法不能保持水体深度与污染物浓度同时为正，难以处理干/湿界面以及污染物浓度接近于零的极端情形，而且针对解的光滑区域不具备高阶精度，此外在强间断附近亦缺乏高分辨属性。近年来兴起的具有保正属性的高精度数值方法为浅水中污染物模型的数值模拟提供了新思路。为了开展高效的数值模拟，本项目将以污染物模型为研究对象，以申请人针对污染物模型的前期研究以及贴体网格的预实验结果为基础，基于非结构网格提出一种改进的有限体积WENO格式，保持一般源项与对流项之间的精确平衡，保持水体深度与污染物浓度同时为正，获得高阶精度与高分辨属性，同时拥有更高的计算效率，更好地满足实际问题的需求。本项目的科学意义在于以新的视角揭示浅水中污染物的运动规律，深入阐明其输运过程的复杂机理，研究成果将进一步丰富计算流体力学领域的理论与方法，同时为水资源保护提供新的思路与技术支持，且在生态保护领域亦有潜在应用前景。

本项目研究针对由于底部不平的浅水中的污染物模型，设计了well-balanced有限差分WENO格式，严格的理论分析与广泛的一、二维数值实验皆表明该格式能够精确保持精确守恒属性，且在强间断附近保持陡峭的间断过渡，此外针对光滑解拥有高阶精度。针对传统的WENO格式计算成本较高的缺点，我们设计了基于间断指示子的杂交WENO格式，数值结果表明，该杂交格式能够节约比较可观的计算成本，同时保持了传统WENO格式的原有良好性质。为了克服WENO格式对于网格要求较高的特点，借助于坐标变换，我们设计了复杂计算区域上的杂交WENO格式, 计算结果非常理想。上述研究结果不仅为计算流体力学领域提供了新的理论与方法，而且为预防环境污染提供了技术支持。此外，在该项目的支持下，我们在本研究领域重要期刊发表SCI论文5篇，指导硕士研究生4人。