School of Science Department of Mathematics 50 Efficient Numerical Methods for Dynamic Interface Supervisor: LEUNG, Shing Yu / MATH Student: WONG, Cheuk Yuen / MATH-GM WONG, Shing Yin / MATH-CS Course: UROP1000, Summer UROP1000, Summer In this report we would discuss our own developed numerical methods for the surface area preserving mean curvature flow. We could achieve the convergence of the curve with preserved arc-length using 25 iteration with condition Dt =0.001, q size of 128. However, we failed to achieve the expected result with the following condition: Dt ³ 0.001, q size of 128, and q size of 600, Dt =0.001. The first failed condition led to short-term convergence of the curve with a non-decreasing arc-length. The second failed condition resulted in a rapid increase in arc-length and divergence of the curve. In this report, we will analyze these failed conditions and provide reasoning for why they did not yield the expected result. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG, Shing Yu / MATH Student: CHEN, Rulin / SSCI Course: UROP1100, Summer The Finite-Time Lyapunov Exponent (FTLE) plays a significant role in the study of fluid dynamics as it provides valuable insights into the convergence and divergence of fluid flow. In this research, our objective is to compute the FTLE in order to analyze the separation of particles on the surface of a sphere (S^2) under a given velocity field. To accomplish this, we employ Runge-Kutta methods to solve the ordinary differential equations (ODEs) involved in computing the FTLE. However, unlike traditional Runge-Kutta methods, we incorporate projection at each time step (h) to ensure that the particles remain on the sphere's surface. This step is crucial for accurate results. In our research, we investigate the relationship between the error E(h) and the order of h to determine the most suitable method for computation. We aim to identify a method that minimizes the error and enhances the accuracy of our results. Finally, to visualize the forward and backward FTLE results, we utilize MATLAB to generate informative figures that depict the outcomes of our analysis. Numerical Methods for Solving PDEs on Surfaces Supervisor: LEUNG, Shing Yu / MATH Student: SU, Yilin Hanako / SSCI Course: UROP1000, Summer Wave phenomena are frequently observed in our physical world, often going unnoticed. They manifest in various forms, such as the vibrations of guitar strings, the oscillations of the stock market, and the modeling of ocean wave currents. Waves play a fundamental role in our daily lives. Furthermore, the advection of our atmosphere contributes to the regulation of the Earth's temperature, which is intimately connected to the global issue of climate change, demanding global attention. This UROP project focused on analyzing the timedependent linear and nonlinear wave equations. By delving into these equations, I aim to gain insights into the behavior and characteristics of waves, advancing our comprehension of this significant area of study.
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