Application Of Discontinuous Galerkin Methods For Time Fractional Order Equations And Fluid Dynamics Equations

The discontinuous Galerkin method for partial differential equations has become an important tool in the study on scientific research areas,such as nature science,engineering techniques and economic administration etc.In this thesis,we apply discontinuous Galerkin(DG)methods for solving the time fractional-order equations and the compressible flow equations.In many fields of science and engineering,including materials,mechanical,and biological systems,the fractional order differential equations have advantages over the classical integer-order partial differential equations in modelling some materials with memory,heterogeneity or inheritable character.We solve the time fractional diffusion equations and the time fractional diffusion-wave equations by using direct discontinuous Galerkin(DDG)methods in chapter 2 and chapter 3.The fractional order equations are discretized by finite difference methods in time and DDG methods in space.The stability and convergence of the fully-discrete schemes are analysed.Numerical examples are shown to illustrate the optimalorder convergence in space for both implicit DDG schemes and explicit DDG schemes,and each has advantages: The implicit DDG scheme needs to solve the global system of algebraic equations,while it is unconditionally stable;although the explicit DDG scheme is conditionally stable,it can remain the property of DG methods that the computation can proceed element by element,so that the computing efficiency is still higher.Compared with local discontinuous Galerkin(LDG)methods,DDG methods do not need to introduce any auxiliary variable,so that the scheme is more simple and the computation cost is reduced.The fluid dynamics equations can be used to model the complex physics problems including the explosion dynamics,inertial confined fusion(ICF),interface instability.These problems have several characteristics containing moving boundary,stronger discontinuous,great distortion,nonlinear and multi-media et.al,such that it is a great challenge to solve them by numerical methods.In chapter 4,we solve the fluid dynamics equations by employing Arbitrary Lagrange-Euler(ALE)discontinuous Galerkin methods,and an ALE discontinuous Galerkin scheme con-taining the grid moving velocity is constructed.If the grid moving velocity equals to the fluid velocity,it is the Lagrangian scheme;if the grid moving velocity equals to zero,it is the Eulerian scheme.In one dimensional,the grid moving velocity is chosen from the reference [89].The method can capture the material interface exactly.In two dimensional,we give a new grid moving velocity by modifying the formulation in the reference [128],such that the new mesh can be obtained from the value of previous time step,and the iteration and interpolation are avoided;moreover,the new mesh is concentrated on neighborhood of the contact discontinuous and the shock wave.In the computation,we take the HLLC numerical flux,and the TVB slope limiting or WENO reconstruct limiting is used to eliminate possible spurious oscillations in the approximate solution.Some single medium and multi medium problems are presented to demonstrate the high-order accuracy and robustness of the schemes.Compared with the Lagrangian method,due to the grid moving velocity is introduced in our scheme,the computation stopped happened in the Lagrangian method when the mesh is distorted will be avoided by choosing suitable grid moving velocity.