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Meshless space-time method to solve two-dimensional wave equation
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Meshless space-time method to solve two-dimensional wave equation

Niranjan Warnakulasooriya Mahaguruge
University of West Florida,
Master of Science (MS), University of West Florida
2020

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Abstract

Meshless methods utilizing Radial Basis Functions(RBF) have been widely used to find numerical solutions for Partial Differential Equations(PDEs). Unlike the other numerical methods, meshless algorithms are significantly simpler to implement as they do not require a mesh in the simulation domain. Rolland L. Hardy, an Iowa State geodesist, was the first to study using RBF for scattered data interpolations in the early 1970s. He introduced his Multiquadric(MQ) RBF, which has been used to obtain numerical solutions for various types of RBF interpolation problems. In addition to that, E. J. Kansa, in the very early 1990s, made the first attempt to extend RBF interpolation to obtain solutions for PDEs. In this thesis, we propose a numerical scheme, which has been based on Kansa's method, to solve time-dependent PDEs. In contrast to already existing methods for solving time-dependent PDEs, our model treats the time variable the same as a spatial variable. However, the accuracy of the RBF numerical methods highly depends on the shape parameter, c, which is associated with the RBF. The value of the c that guarantees the highest accuracy is problem dependent and it is called as the optimal value of c. Even with the optimal value of c, it is not possible to achieve a significantly high accuracy compared to existing methods. In order to enhance the level of accuracy, we introduce \Ghost Points" into the computational domain. While traditional RBF based numerical methods place the centers exclusively inside the computational domain, the ghost point approach expands the region of the centers inside and outside the computational domain. Our numerical results suggest that the accuracy of the numerical results has been significantly increased by the ghost points.
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