Addressing the challenges of implementation of high-order finite volume schemes for atmospheric dynamics of unstructured meshes

The solution of the non-hydrostatic compressible Euler equations using Weighted Essentially Non-Oscillatory (WENO) schemes in two and three-dimensional unstructured meshes, is presented. Their key characteristics are their simplicity; accuracy; robustness; non-oscillatory properties; versatility in handling any type of grid topology; computational and parallel efficiency. Their defining characteristic is a non-linear combination of a series of high-order reconstruction polynomials arising from a series of reconstruction stencils. In the present study an explicit TVD Runge-Kutta 3rd -order method is employed due to its lower computational resources requirement compared to implicit type time advancement methods. The WENO schemes (up to 5th -order) are applied to the two dimensional and three dimensional test cases: a 2D rising

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

Low-Mach number treatment for Finite-Volume schemes on unstructured meshes

The paper presents a low-Mach number (LM) treatment technique for high-order, Finite-Volume (FV) schemes for the Euler and the compressible Navier–Stokes equations. We concentrate our efforts on the implementation of the LM treatment for the unstructured mesh framework, both in two and three spatial dimensions, and highlight the key differences compared with the method for structured grids. The main scope of the LM technique is to at least maintain the accuracy of low speed regions without introducing artefacts and hampering the global solution and stability of the numerical scheme. Two families of spatial schemes are considered within the k-exact FV framework: the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the Weighted Essentially Non-Oscillatory (WENO). The simulations are advanced in time with an explicit third-order Strong Stability Preserving (SSP) Runge–Kutta method. Several flow problems are considered for inviscid and turbulent flows where the obtained solutions are compared with referenced data. The associated benefits of the method are analysed in terms of overall accuracy, dissipation characteristics, order of scheme, spatial resolution and grid composition

A high-order finite-volume method for atmospheric flows on unstructured grids

This paper presents an extension of a Weighted Essentially Non-Oscillatory (WENO) type schemes for the compressible Euler equations on unstructured meshes for stratified atmospheric flows. The schemes could be extended for regional and global climate models dynamical cores. Their potential lies in their simplicity; accuracy; robustness; non-oscillatory properties; versatility in handling any type of grid topology; computational and parallel efficiency. Their defining characteristic is a non-linear combination of a series of high-order reconstruction polynomials arising from a series of reconstruction stencils. In the present study an explicit Strong Stability Preserving (SSP) Runge-Kutta 3rd-order method is employed for time advancement. The WENO schemes (up to 5th-order) are applied to the two dimensional and three dimensional test cases: a 2D rising thermal bubble; the 2D density current and the 3D Robert smooth bubble. The parallel performance of the schemes in terms of scalability and efficiency is also assessed

A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids

A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the compressible Euler equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The developed method is used to compute a variety of both steady-state and time-accurate flow problems on arbitrary grids. The numerical results demonstrated the superior accuracy of this discontinuous Galerkin method in comparison with a second order finite volume method and a third-order WENO method, indicating its promise and potential to become not just a competitive but simply a superior approach than its finite volume and ENO/WENO counterparts for solving flow problems of scientific and industrial interest

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 New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

International audienceIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order De- tection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decre- menting processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic par- tial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements

Very high-order methods for 3D arbitrary unstructured grids

Understanding the motion of fluids is crucial for the development and analysis of new designs and processes in science and engineering. Unstructured meshes are used in this context since they allow the analysis of the behaviour of complicated geometries and configurations that characterise the designs of engineering structures today. The existing numerical methods developed for unstructured meshes suffer from poor computational efficiency, and their applicability is not universal for any type of unstructured meshes. High-resolution high-order accurate numerical methods are required for obtaining a reasonable guarantee of physically meaningful results and to be able to accurately resolve complicated flow phenomena that occur in a number of processes, such as resolving turbulent flows, for direct numerical simulation of Navier-Stokes equations, acoustics etc. The aim of this research project is to establish and implement universal, high-resolution, very high-order, non-oscillatory finite-volume methods for 3D unstructured meshes. A new class of linear and WENO schemes of very high-order of accuracy (5 th ) has been developed. The key element of this approach is a high-order reconstruction process that can be applied to any type of meshes. The linear schemes which are suited for problems with smooth solutions, employ a single reconstruction polynomial obtained from a close spatial proximity. In the WENO schemes the reconstruction polynomials, arising from different topological regions, are non-linearly combined to provide high-order of accuracy and shock capturing features. The performance of the developed schemes in terms of accuracy, non-oscillatory behaviour and flexibility to handle any type of 3D unstructured meshes has been assessed in a series of test problems. The linear and WENO schemes presented achieve very high-order of accuracy (5 th ). This is the first class of WENO schemes in the finite volume context that possess highorder of accuracy and robust non-oscillatory behaviour for any type of unstructured meshes. The schemes have been employed in a newly developed 3D unstructured solver (UCNS3D). UCNS3D utilises unstructured grids consisted of tetrahedrals, pyramids, prisms and hexahedral elements and has been parallelised using the MPI framework. The high parallel efficiency achieved enables the large scale computations required for the analysis of new designs and processes in science and engineering.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes