Stabilized Seventh-order Dissipative Compact Finite Difference Scheme Using SATs

High-order methods(methods with convergence rate higher than second order)are more efficient than low-order methods since they allow a reduction in the degrees of freedom for certain error span.When applied to the simulation of complex flow fields inside a bounded domain,direct numerical simulation(DNS)and large eddy simulation(LES)of turbulence and aeroacoustic,for instance,high-order methods have better resolution and computational efficiency.Specifically,high-order finite difference methods are ideal for problems with broadband length scales.However,the major difficulty of deriving high-order finite difference scheme is the construction of stable boundary closures.It is necessary for long-time simulation that the numerical error of the scheme does not diverge as time increases.The main work of this thesis is to modify a seventh-order dissipative compact scheme so that it could achieve optimal globally seventh-order accuracy,and to introduce the so-called simultaneously approximation term method to guarantee the time stability of the scheme.In this thesis a new interpolation scheme and difference scheme near boundary will be constructed,resulting in a globally seventh-order scheme.Then the simultaneously approximation term method will be introduced to weakly impose boundary condition,and eigenvalue analysis will be carried out to study the time stability of the obtained scheme.The scheme will be extended to non-linear case by taking the Euler equations as example,and validated by relevant numerical tests.Afterwards,a two-dimensional interface treatment technique via simultaneously approximation term method will be presented.Numerical test on two-dimensional vortex convection problem verified the validity of the technique.Finally,in order to apply the scheme to simulation of realistic problems,the case with curvilinear grid,especially the implementation of simultaneously approximation term method in this case,is studied.By using the simultaneously approximation term method to impose slip-wall boundary condition,numerical examples of two-dimensional nozzle flow and two-dimensional flow with a bum are performed to validate the scheme.