Research On The High Order Accurate Numerical Methods On Unstructured Grids

Unstructured grids play an important role in computation fluid dynamics for theflexibility in handling complex geometries, and the high order numerical methods aresignificant to improve computational efficiency. However, since the unstructuredhigh-order accurate methods are rather complicated and still have many issues toaddress, currently they are not so widely used. This thesis focuses on the developmentof high order finite volume methods and discontinuous Galerkin methods, and tries tosolve some importrant and common problems in high order methods, such as theshock-capturing strategy and curved boundary treatment, then applies these methods insolving the compressible flows.As an extension of the widely-used second-order finite volume (FV) method, thehigh-order finite volume methods in the present paper are based on the so-called k-exactreconstruction, which construct the high order polynomial distribution inside a cell. Thisthesis mainly solves the following problems which have significant influences on theefficiency and robustness of the high-order FV methods. The first is the high-orderdiscretization for the viscous terms and the viscous boundary treatment. This thesisgives a flux-averaged method for the viscous terms and constrained least-square methodfor the viscous boundary treatment. The second is the design of the high order limitersto suppress the numerical oscillations near the discontinuities and to assure the highorder accuracy in smooth region. This thesis proposes a simple yet efficient secondaryreconstruction (SR) procedure for providing the candidate polynomials used in thelimiting procedure, and develops three types of limiters, namely the quadrature-freek-exact WENO limiters, WBAP limiters and successive weighted limiters, based on theSR procedure. And numerical examples with strong shocks are used to validate theefficiency of these limiters. The third is the generalization of the high order FV methodsto mixed elements cases and problems with curved boundary. The iso-parametrictransformation is used to transform the mixed cells and cells with curved boundary toregular region, thus the high order numerical integration can be carried out. Throughthese three steps, the FV methods can be applied in the simulations ofrelatively-complex compressible flows, and the results show the higher efficiency for the higher order schemes.Also the successful strategies in FV methods are generalized to thediscontinuous Galerkin (DG) methods. Firstly we general the k-exact WENOlimiters as well as the SR to DG methods, which preserve the compactness andhigh order accuracy of DG method and suppress the numerical oscillations nearthe shocks. Secondly the treatment for curved boundary is applied in DGmethods to keep the high order accuracy in boundary cells. Abundance ofnumerical examples also validate the efficiency and superiority of the DGmethods.