This project undertakes a comprehensive study of the linear advection equation, focusing on the accuracy, stability, and convergence of various numerical solution methods. The methodologies employed include Forward Time Backward Space (FTBS), Forward Time Forward Space (FTFS), Forward Time Center Space (FTCS), Lax, Lax-Wendroff, Leap-Frog, Θ-method, and High-order 4-2 schemes. The project investigates the theoretical accuracy and stability of each algorithm, supported by numerical tests and comparisons between analytical and numerical results. Boundary conditions and initial conditions, including square and Gaussian waves, are carefully considered. The discussion encompasses stability analysis, including the Courant number's impact, and the application of Fourier analysis and spectral methods. The results, presented through MATLAB-generated figures, illustrate the behavior of different methods under various conditions, offering insights into their performance. The conclusion synthesizes the findings regarding the accuracy, stability, and convergence of the linear advection equation's numerical solutions.
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