Advancements in Meshfree Methods for Hyperbolic Conservation Laws on High-Dimensional Manifolds

Hyperbolic conservation laws on higher-dimensional forms are crucial in understanding various physical phenomena. These mathematical structures help us describe and predict the behavior of different systems, such as traffic flow, fluid dynamics, and wave propagation. Improving our knowledge and computational methods in this field could lead to advancements in engineering, meteorology, and physics, which is the core motivation for our project.

We begin our research by generalizing an embedding method used for solving hyperbolic conservation laws on 1D manifolds. One significant challenge we face is understanding how hyperbolic characteristics interact with the manifold’s shape and the conservative variables’ small- and large-scale structures while still maintaining desirable properties. Next, we plan to develop an alternative solver based on an intrinsic formulation. The goal is to reduce computational complexity by working in the parameter space. However, this necessitates working with patches of the manifold. Our first objective is to adapt essentially non-oscillatory schemes to work with data points stored in patches. We will explore how the placement of high-gradient and fine-scale solutions across these patches impacts our solutions’ accuracy and numerical stability. We will then identify an effective data structure for storing the solutions of hyperbolic conservation laws in the local parameter space.

Recognizing that no real-life surface is perfectly smooth and that roughness is a common feature in natural and man-made environments, our final research plan focuses on examining how surfaces with inherent roughness affect the solutions of scalar hyperbolic conservation laws. We aim to test the effectiveness of our developed embedding and intrinsic methods in capturing these changes and gain insights into hyperbolic solutions’ behaviors as the roughness amplitude increases. By simulating more complex hyperbolic conservation laws, such as Euler, Navier-Stokes, and shallow water-wave equations, on rough manifolds, we expect our results to provide a more grounded and reliable understanding of the physical phenomena under study. These simulations will also serve to test our models and enhance their ability to simulate outcomes that align with real-world expectations and fundamental physics principles.