Numerical Methods for Solving Hyperbolic Type Problems

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Abd Al-Haq, Anwar Jamal Mohammad
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An-Najah National University
Hyperbolic Partial Differential Equations play a very important role in science, technology and arise very frequently in physical applications as models of waves. Hyperbolic linear partial differential equations of second order like wave equations are the ones to be considered. In fact, most of these physical problems are very difficult to solve analytically. Instead, they can be solved numerically using some computational methods . In this thesis, homogeneous and inhomogeneous wave equations with different types of boundary conditions will be solved numerically using the finite difference method (FDM) and the finite element method (FEM) to approximate the analytical (exact) solution of hyperbolic PDEs. The discretizing procedure transforms the boundary value problem into a linear system of n algebraic equations that can be solved by iterative methods. These iterative methods are: Jacobi, Gauss-Seidel, SOR, and Conjugate Gradient methods. A comparison between these iterative schemes is drawn. The numerical results show that the finite difference method is more efficient than the finite element method for regular domains, while the finite element method is more accurate for complex and irregular domains. Moreover, we observe that the Conjugate Gradient iterative technique gives the most efficient results among the other iterative methods.