Bernstein-Bezier Techniques for High Order Time-Domain Discontinuous Galerkin Methods

波传播的模拟是许多不同领域应用的支柱，如医学成像、预测地震学和工程设计。许多应用需要在具有复杂边界或特征的域中重复进行高分辨率的波模拟。例如，脑成像方法需要处理通过人体内变化介质的波传播，例如不同类型组织之间的转换或从组织到骨骼的转换。目标应用领域需要可靠、准确、高效的数值模拟。然而，结合复杂介质和几何形状的精确近似通常会牺牲这些属性中的一个或多个，导致模拟准确但缓慢，快速但不准确，或者在特定设置之外不稳定。拟议的项目旨在开发可靠的、可证明稳定的方法，以保持波在复杂介质和几何形状中的传播精度和效率。该项目将使用高阶不连续伽辽金（DG）方法研究数值波传播，主要包括两个任务。第一个任务将利用高性能架构，如图形处理单元（gpu），以及最近开发的权重调整和Bernstein-Bezier DG方法。权重调整DG方法在变量介质存在时具有可证明的稳定性和高阶精度，但在近似阶数方面具有较高的计算复杂度。Bernstein-Bezier DG方法在顺序方面实现了最佳的计算复杂度，但需要低分辨率的介质和几何模型才能做到这一点。第一项任务是将这两种方法结合起来，构建稳定、高阶精度和低计算复杂度的波在不同介质中的传播方法。第二个任务将是提高使用曲线非结构化网格建模复杂边界的可靠性。典型的曲线网格生成方法会产生不适合数值模拟的单元。第二项任务将寻求通过利用多项式的伯恩斯坦-贝塞尔表示来构建高阶几何近似的鲁棒方法，这与底层函数的形状属性密切相关。

Simulations of wave propagation are the backbone of numerous applications in a diverse set of areas, such as medical imaging, predictive seismology, and engineering design. Many applications require repeated high resolution simulations of waves in domains with complex boundaries or features. For example, brain imaging methods require dealing with wave propagation through changing media within the human body, such as transitions between different types of tissue or transitions from tissue to bone. The target application areas require numerical simulations which are reliable, accurate, and efficient. However, incorporating accurate approximations of complex media and geometries typically sacrifices one or more of these properties, resulting in simulations which are accurate but slow, fast but inaccurate, or unstable outside of specific settings. The proposed project aims to develop reliable, provably stable methods for wave propagation in complex media and geometries which retain accuracy and efficiency. The project will study numerical wave propagation using high order discontinuous Galerkin (DG) methods, and consists of two main tasks. The first task will leverage high performance architectures such as Graphics Processing Units (GPUs), as well as the recently developed weight-adjusted and Bernstein-Bezier DG methods. Weight-adjusted DG methods achieve provable stability and high order accuracy in the presence of variable media, but have a high computational complexity with respect to the order of approximation. Bernstein-Bezier DG methods achieve an optimal computational complexity with respect to the order, but require low-resolution models of media and geometry in order to do so. The first task will combine these two approaches to construct methods for wave propagation in varying media which are stable, high order accurate, and have low computational complexity. The second task will be to improve the reliability of modeling complex boundaries using curvilinear unstructured meshes. Typical methods for producing curvilinear meshes can result in elements unsuitable for numerical simulation. The second task will seek robust methods for constructing high order approximations of geometry by leveraging Bernstein-Bezier representations of polynomials, which are closely related to shape properties of the underlying function.