This thesis presents a methodology for the numerical solution of one-dimensional (1D) and
two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation.
The 1D Euler equations are used to assess the performance and stability of the discretisation.
The explicit time restriction is derived and it is established that the optimal
polynomial degree, p, in terms of efficiency and accuracy of the simulation is p = 5. Since
the method is characterised by minimal diffusion, it is particularly well suited for the
simulation of the pressure wave generated by train entering a tunnel. A novel treatment
of the area-averaged Euler equations is proposed to eliminate oscillations generated by the
projection of a moving area on a fixed mesh and the computational results are validated
against experimental data.
Attention is then focussed on the development of a 2D DG method implemented
using the high-order library Nektar++. An Euler and a laminar Navier–Stokes solvers
are presented and benchmark tests are used to assess their accuracy and performance. An
artificial diffusion term is implemented to stabilise the solution of the Euler equations in
transonic flow with discontinuities. To speed up the convergence of the explicit method,
a new automatic polynomial adaptive procedure (p-adaption) and a new zonal solver are
proposed. The p-adaptive procedure uses a discontinuity sensor, originally developed as
an artificial diffusion sensor, to assign appropriate polynomial degrees to each element
of the domain. The zonal solver uses a modification of a method for matching viscous
subdomains to set the interface conditions between viscous and inviscid subdomains that
ensures stability of the flow computation. Both the p-adaption and the zonal solver
maintain the high-order accuracy of the DG method while reducing the computational
cost of the simulation
