Study On Fast And Parallel Algorithm For Differential Dynamics Equations

With the rapid development of computer technology and numerical analysis theory, the high accuracy method for solving complex flow problems is one of the most important issue of the computational fluid dynamics. The complex problems encountered in engineering problems often require computing performance of the hardware to hundreds of billions or even trillions, especially high accuracy algorithms on unstructured grids existed some shortcomings, such as more calculation time-consuming and large storage space than low accuracy algorithms. The robustness of higher order schemes is worse than the lower order one, and the higher order algorithms are more complex than the lower order one, which make the calculation cost sharply increase, so the higher order algorithms have not been used widely in engineering problems. However the superior geometry flexibility and good universality of unstructured grids can make the grids adaptive process. Then the unstructured grids is suitable for handling complex boundary problems. So the high performance parallel computing has become an important way of scientific computation on the unstructured grids.In the present paper, the high-performance numerical finite volume method, discontinuous Galerkin finite element method, implicit algorithms and parallel algorithms for compressible Euler equations are carried out on unstructured grids systematically. In addition, the high-performance parallel numerical algorithm for compressible Navier-Stokes equations is given on unstructured grids. At last, the alternating direction parallel algorithm with parameters for solving large banded linear systems is presented.In this paper, it mainly includes the following contents:1. The space discrete schemes for compressible Euler equations and compressible Navier-Stokes equtions are studied including finite volume method and disco ntinuous Galerkin method from the physical significance and practical calculation condition of the flow. The numerical fluxes determine the soution accuracy of computing problems and the computational results. Therefore, the impact of different numerical flux, different calculation limiter is illustrated for computational results. Finally, the numerical experiments indicate that FVM and DGM have advange of efficiency and effectiveness.2. In this part, the local discontinuous Galerkin format and discontinuo us Galerkin scheme are discussed, and the differences of discontinuous Galerkin method and local discontinuous Galerkin method are analyzed for solving Burgers equations. The two-dimensional Riemann problem is solved by limiter based on generalized Fourierideas. The results are analyzed for the various states flow field, which show the feasibility of discontinuous Galerkin method. Finally, the examples are calculated with different numerical flux. The local discontinuous Galerkin method is adopted to solve the laminar flat problem. The results verify the feasibility and efficiency of local discontinuous Galerkin method, which coincide with Blasius solutions.3. The SO R inner iterative LU-SGS scheme is adopted to solve the Euler equations, which based on eigenvalues splitting technique and make the CFL stability condition significantly weak, then the ODEs is changed the large sparse linear systems. So the implicit algorithms can improve computational efficiency. Finally, the numerical experiments implemented on the flows around NACA0012, RAE2822 airfoil and ONERA M6 verify the efficient performance of the implicit algorithm. N umerical results show that the performance of the present algorithm is much better than the traditional LU-SGS algorithm one, the computation efficiency reached more than 3 times, close to the GMRES algorithm.4. In order to further improve computational efficiency, the serial algorithm has been unable to meet the computing needs, the parallel FVM on unstructured grids is proposed for inviscid compressible Euler equations. For parallel computing, it is very important that the definition of virtual grid and information transfer among local grids. So this paper proposes the virtual grid technique to ensure load balancing, which ensure that the load among the processors are basically the same. The informations transfer between processors, boundary treatment and parallel implementation process are given. The numerical results and parallel performance are analyzed. Finally, the flows around NACA0012 airfoil and ONERA M6 wing are calculated, the results verify that the present parallel algorithm is feasible.5. In order to further improve computational efficiency, the parallel FVM on unstructured grids is proposed for Navier-Stokes equations with SA turbulance model and GMRES scheme. For different grid types, this paper proposes the virtual grid technique to ensure load balancing and reduce the waiting time, which ensure that the load among the processors are basically the same. The numerical results and parallel performance are analyzed. Finally, the flows around NACA0012 airfoil and RAE28822 airfoil are calculated, the results verify that the present parallel algorithm is feasible.6. An effcient parallel iterative method with parameters on distributed- memory multicomputer is investigated for solving the banded linear equations in this work. The parallel algorithm at each iterative step is executed using alternating direction by splittingthe coeiffcient matrix and using parameters properly. Only it twice requires the communications of the algorithm between the adjacent processors, so this method has high parallel effciency. Some convergence theorems for diferent coe ffcient matrices are given, such as a Hermite positive deinite matrix or an M- matrix. N umerical experiments implemented on cluster verify that the present algorithm has the advantages over the others classical iterative algorithms of high effciency and low memory space.