Application Of Discontinuous Galerkin Method For Convection-diffusion Equstions And Incompressible Flow

Convection-diffusion equation and incompressible NS equations arevery important basic equations in fluid dynamics and incompressible flow.They can be applied in naval architecture and ocean engineering,environmental science, development of new energy resources, machinerydesign and manufacture, electronic science and etc. Research aboutnumerical method of these equations is absolutely meaningful.Convection dominated problems with strong discontinuity are alsocommon problems in fluid dynamics, for example, shock wave problem,tow phase flow problem and etc. Traditional numerical methods cannotsolve these problem well，because there will be numerical dissipation anddispersion in these method. So when apply these method to solveconvection-dominated problems with strong discontinuity, high orderaccuracy is hard to be achieve and the result is easy to become distorted.Discontinuous Galerkin method have advantages of both Finite Volumemethod and Finite Element method, this method can use high order shapefunction and the solution space of discontinuous Galerkin method is notcontinuous. So it is very suitable for problems with strong discontinuity. Forconvection dominated problem, discontinuous Galerkin method can fullyuse the physical feature of convection problem, can get high accuracy andhave less numerical dissipation. This method is also suitable for convectiondominated problem too.In this paper, convection-diffusion equation and incompressible NSequations will be chosen. After research the method, the LDG solver, HDGsolver and BR2-DG solver for convection-diffusion equation and incompressible NS equations will be developed independently. Theaccuracy, stabilization and convergence will be analyzed through someexamples. In general, if applying discontinuous Galerkin method to solveconvection-diffusion equation, higher accuracy can be achieved by usinghigher order shape function or better grid. The result and local conservationis very good for convection-diffusion equation, even for convectiondominated problem with discontinuous boundary condition or initialcondition. When applying HDG method to solve incompressible NSequations, both velocity and pressure can get high accuracy andconservation is very good in HDG method. In contrast, BR2-DG method ismore efficient and stable. Discontinuous Galerkin method have boardprospects and development potential.