Numerical simulation of shock propagation in one and two dimensional domains
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The objective of this dissertation is to develop robust and accurate numerical methods for solving the compressible, non-linear Euler equations of gas dynamics in one and two space dimensions. In theory, solutions of the Euler equations can display various characteristics including shock waves, rarefaction waves and contact discontinuities. To capture these features correctly, highly accurate numerical schemes are designed. In this thesis, two different projects have been studied to show the accuracy and utility of these numerical schemes. Firstly, the compressible, non-linear Euler equations of gas dynamics in one space dimension are considered. Since the non-linear partial differential equations (PDEs) can develop discontinuities (shock waves), the numerical code is designed to obtain stable numerical solutions of the Euler equations in the presence of shocks. Discontinuous solutions are defined in a weak sense, which means that there are many different solutions of the initial value problems of PDEs. To choose the physically relevant solution among the others, the entropy condition was applied to the problem. This condition is then used to derive a bound on the solution in order to satisfy L2-stability. Also, it provides information on how to add an adequate amount of diffusion to smooth the numerical shock waves. Furthermore, numerical solutions are obtained using far-field and no penetration (wall) boundary conditions. Grid interfaces were also included in these numerical computations. Secondly, the two dimensional compressible, non-linear Euler equations are considered. These equations are used to obtain numerical solutions for compressible ow in a shock tube with a 90° circular bend for two channels of different curvatures. The cell centered finite volume numerical scheme is employed to achieve these numerical solutions. The accuracy of this numerical scheme is tested using two different methods. In the first method, manufactured solutions are used to the test the convergence rate of the code. Then, Sod's shock tube test case is implemented into the numerical code to show the correctness of the code in both ow directions. The numerical method is then used to obtain numerical solutions which are compared with experimental data available in the literature. It is found that the numerical solutions are in a good agreement with these experimental results.