The Application Of Post-processing Technique For The Discontinuous Galerkin Method For Aeroacoustic Problems

Computational Aero-Acoustics（CAA）is an interdisciplinary subject involving acous-tics,fluid mechanics,numerical analysis,computational geometry,computer science and partial differential equations.CAA not only uses numerical methods to study fluid noise and the mechanism of fluid noise interaction with objects,but also uses these numerical methods to study practical problems in aeroacoustics,providing tools for controlling and reducing aeroacoustic noise.So CAA has always been an active research area.Among many CAA methods,the discontinuous Galerkin（DG）method,which has the advantages of h-p adaptability,has been widely used in solving aeroacoustics problems.Due to the high precision requirements of numerical solutions in practical problems,post-processing technology is applied in this thesis to extract the“hidden precision”information of DG solutions at the last moment of solving aeroacoustics problems,so as to improve the ac-curacy of DG solutions.In this thesis,we mainly study how to improve the accuracy of DG solutions with Smoothness-Increasing Accuracy-Conserving（SIAC）filter for aeroacoustics problems.This thesis firstly introduces the basic knowledge required for theoretical analysis,and then gives the linearization process of Euler system by taking one-dimensional Euler sys-tem as an example,and deduces the exact solution of linear Euler system under the acoustic pulse generated by the initial Gaussian pressure distribution in the center of the computing domain.Secondly,for the matrix-valued functions A₁and A₂,variable coefficients hy-perbolic conservation laws system（2-1）can be used by the same matrix diagonalization.When the numerical flux is Lax-Friedrichs fluxes,The k+1 estimate of the difference of DG error in L²norm is given according to the properties of the discrete operator of DG.The 2k+1 negative norm error estimate of difference is given by using the L²norm estimate and the duality problem,and it is proved that the post-processing solution can reach the 2k+1 superconvergence.The idea of proof in this thesis is to start with the one-dimensional hyperbolic conservation laws system,and then finally extend to two di-mensions.Finally,this thesis shows the L²errors and convergence orders of different numerical examples before and after post-processing,and plots the error diagrams of nu-merical examples,and verifies that the accuracy of DG solution improved to 2k+1 by SIAC filtering.In this thesis,the superconvergence analysis of the post-processing solutions is ob-tained,which is an appropriate supplement to the post-processing technical theory in CAA,and the negative norm estimation of DG method for the system of hyperbolic conservation laws with variable coefficients is given in the proof process.