Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes

This is the continuation of the paper "central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction" by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to nite volume schemes on non-staggered grids. This takes a new nite volume approach for approximating non-smooth solutions. A critical step for high order nite volume schemes is to reconstruct a nonoscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th order schemes without characteristic decomposition.The research of Y. Liu was supported in part by NSF grant DMS-0511815. The research of C.-W. Shu was supported in part by the Chinese Academy of Sciences while this author was visiting the University of Science and Technology of China (grant 2004-1-8) and the Institute of Computational Mathematics and Scienti c/Engineering Computing. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345. The research of E. Tadmor was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076. The research of M. Zhang was supported in part by the Chinese Academy of Sciences grant 2004-1-8

A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids

A reconstructed discontinuous Galerkin (RDG) method based on a hierarchical WENO reconstruction, termed HWENO (P1P2) in this paper, designed not only to enhance the accuracy of discontinuous Galerkin methods but also to ensure the nonlinear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this HWENO (P1P2) method, a quadratic polynomial solution (P-2) is first reconstructed using a Hermite WENO reconstruction from the underlying linear polynomial (P-1) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the RDG method. The developed HWENO (P1P2) method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, robustness, and non-oscillatory property. The numerical experiments indicate that the HWENO (P1P2) method is able to capture shock waves within one cell without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method

Multi-dimensional Limiting Strategy for Higher-order CFD Methods - Progress and Issue (Invited)

The present paper deals with the progress of multi-dimensional limiting process (MLP) and discuss the issues for further improvements. MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillationcontrol mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn approximation or reconstruction. The resulting algorithm, called the hierarchical MLP, facilitates the capturing of detailed flow structures while maintaining the formal order-of-accuracy in smooth region and providing accurate non-oscillatory solutions across discontinuous region. This algorithm has been developed within the modal DG framework, but it also can be formulated into a nodal framework, most notably the CPR framework. Troubled-cells are detected by applying the MLP concept, and the final accuracy is determined by the projection procedure and the hierarchical MLP limiting step. Through extensive numerical analyses and computations ranging from scalar conservation laws to fluid systems, it is demonstrated that the proposed limiting approach yields the outstanding performances in capturing compressible inviscid and viscous flow features. Further issues are also mentioned to improve and extend the current approach for higher-order simulations of high-Reynolds number compressible flows.Authors appreciate the financial supports by the EDISON program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2011-0020559) and by NSL (National Space Laboratory) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2014M1A3A3A02034856). This work is also partially supported by the RoK ST&R project of Lockheed Martin Corporation. Authors also appreciate the computing resources provided by the KISTI Supercomputing Center(KSC-2014-C3-054).OAIID:RECH_ACHV_DSTSH_NO:420150000004648007RECH_ACHV_FG:RR00200003ADJUST_YN:EMP_ID:A001138CITE_RATE:FILENAME:6.2015-3199.pdfDEPT_NM:기계항공공학부EMAIL:[email protected]_YN:FILEURL:https://srnd.snu.ac.kr/eXrepEIR/fws/file/a984d649-4b23-435b-adc9-df9aa0c8aa46/linkCONFIRM:

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

This is the published version. Copyright 2011 © EDP SciencesThe newly developed unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be completely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compressible Navier-Stokes equations to demonstrate its performance