| Abstract |
In this thesis, I present the Surface Discretization-Corrected Particle Strength Exchange (Surface DC-PSE) method for vector-valued partial differential equations (PDEs) on curved surfaces. Surface DC-PSE is a numerical meshfree collocation method for approximating surface differential operators. The method requires a surface point cloud and the surface normal at each point, and provides a mathematically embedding but computationally embedding-free approximation of the fields and their spatial derivatives at each point. The Surface DC-PSE method is an extension of the DC-PSE method, which is a mesh- free numerical method for approximating Euclidean differential operators. Surface DC-PSE is based on the idea of extending the field constant along the normal into the embedding space, constructing the kernels using both surface and “virtual” extended particles along the normal, and collapsing the resulting ”bulk” kernels into surface ones. This thesis presents a comprehensive study of the Surface DC-PSE method, including its mathematical formulation, numerical implementation, and verification tests. The results show that the method provides convergent and stable approximations of scalar and vector surface differential operators, and is applicable to a wide range of surfaces. I also discuss the limitations of the method, including the requirement of a homogeneous point cloud, the sensitivity to particle distributions, and the computational cost of constructing the kernels. Future work is proposed to address these limitations, including modifying the neighborhood construction algorithm, evaluating the accuracy and stability of vector Surface DC-PSE on deforming surfaces, and exploring applications of the method to non-analytical surfaces. The Surface DC-PSE method has the potential to be used in a wide range of applications, including simulating biological fluid surfaces, studying morphogenesis, and modeling complex surface phenomena. The method provides a robust and reliable framework for approximating surface differential operators, and it provides a basis for further research and development in these and other areas of scientific computing. |