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By Wanai Li

This thesis makes a speciality of the advance of high-order finite quantity equipment and discontinuous Galerkin tools, and provides attainable recommendations to a couple of very important and customary difficulties encountered in high-order tools, equivalent to the shock-capturing approach and curved boundary remedy, then applies those easy methods to remedy compressible flows.

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I=−∗ It is easy to know that the scheme satisfying TVD condition is also monotonicity preserving. Essentially Non-Oscillatory (ENO) [2]. It means in the high-order reconstruction of smooth function V (x), the piecewise smooth polynomial P(x) is obtained which achieve high-order accuracy in smooth regions and avoid the Gibbs phenomenon near the discontinuities. In fact, ENO is high-order limiting strategy, while the TVD scheme is at most second order accurate [3]. Thus, the ENO scheme does not satisfy the TVD condition.

Other types of limiters such as the TVB limiter [9], Hermite WENO limiter [10], and moment limiter [11] are also adopted in the DGM and SV/SD methods. Summarizing on the successful high-order limiters, we find the three components in the limiters, namely the construction of candidate polynomials, the limiting function and the limiting process. Two classical high-order limiters are analyzed in terms of these three components in the below. The WENO limiter uses the multiple 40 3 Accuracy Preserving Limiters for High-Order Finite Volume Methods reconstructions to provide the candidate polynomials, the smoothness indicator to compute the WENO weight as the limiting function and the weighted summation for the limiting process.

5 Parallelization for Large-Scale Computation In the practical engineering applications, large-scale grids are required to make the calculation more accurate, which always need to work on several computers simultaneously. In the computation, we need to divide the computational grids into partitions and one process is responsible for one partition. The most challenge is the data exchange in the interfaces of partitions. Comparing to the second-order scheme, the parallelization for the high-order FVM is more difficult since we need to create a 26 2 High-Order Finite Volume Method for the Compressible Flows Partition 1 Partition 2 Partition 3 Adjacent to 2 Adjacent to 3 Adjacent to 1 Adjacent to 3 Adjacent to 1 Adjacent to 2 Fig.

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