|dc.description.abstract||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
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.||en