Yidong Xia

OpenACC-based GPU Acceleration of a 3-D Unstructured Discontinuous Galerkin Method

A GPU-accelerated discontinuous Galerkin (DG) method is presented for the solution of compressible flows on 3-D unstructured grids. The present work has employed two of the most attractive features in a new programming standard of parallel computing – OpenACC: 1) multi-platform/compiler support and 2) descriptive directive interface to upgrade a legacy CFD solver with the capacity of GPU computing, without significant extra cost in recoding, resulting in a highly portable and extensible GPU-accelerated code. In addition, a face renumbering/grouping scheme is proposed to overcome the “race condition” in facebased flux calculations that occurs on GPU vectorization. Performance of the developed double-precision solver is assessed for both simple and complex geometries. Speedup factors up to but not limited to 24× and 1.6× were achieved by comparing the measured computing time of the OpenACC program running on an NVIDIA Tesla K20c GPU to that of the equivalent MPI program running on one single core and full sixteen cores of an AMD Opteron-6128 CPU respectively, indicating a great potential to port more features of the underlying DG solver into the OpenACC framework.

An Implicit Method for a Reconstructed Discontinuous Galerkin Method on Tetrahedron Grids

An implicit method for a reconstructed discontinuous Galerkin (RDG) method is presented to solve compressible flow problems on tetrahedron grids. The idea is to combine the accuracy of the RDG method and the efficiency of implicit methods to obtain a better numerical algorithm in computational fluid dynamics. A least-squares reconstruction method is presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via an in-cell reconstruction process. The devised in-cell reconstruction is able to augment the accuracy of the DG method by increasing the order of the underlying polynomial solution. A matrix-free GMRES (generalized minimum residual) algorithm with an LU-SGS (lower-upper symmetric Gauss- Seidel) preconditioner is presented to solve an approximate system of linear equations arising from the Newton linearization. The implicit method is used to compute a variety of three-dimensional problems on tetrahedron grids to assess its accuracy and robustness. The numerical experiments demonstrate that the implicit reconstructed discontinuous Galerkin method can obtain an overall speedup of more than two orders of magnitude for all test cases compared with multi-stage Runge-Kutta reconstructed DG methods. The numerical results also indicate that this implicit RDG(P1P2) method can deliver the desired third-order accuracy, while maintaining advantage in cost over the implicit DG(P2) method.

A WENO Reconstruction-Based Discontinuous Galerkin Method for Compressible Flows on Hybrid Grids

A WENO reconstruction-based discontinuous Galerkin method RDG(P1P2), designed not only to enhance the accuracy of discontinuous Galerkin method but also to ensure linear stability of the RDG method, is presented for solving compressible flow problems on hybrid grids. In this RDG(P1P2) method, a quadratic polynomial solution (P2) is first reconstructed using a least-squares method from the underlying linear polynomial (P1) discontinuous Galerkin solution. 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 and consistent with the underlying DG method. The final quadratic polynomial solution is then obtained using a WENO reconstruction, which is necessary to ensure linear stability of the RDG method. The developed RDG method is used to compute a variety of flow problems on hybrid meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments demonstrate that the developed RDG(P1P2) method is able to maintain the linear stability, achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method without significant increase in computing costs and storage requirements.

An implicit, reconstructed discontinuous Galerkin method for the unsteady compressible Navier-Stokes equations on 3D hybrid grids

An implicit, third-order, reconstructed discontinuous Galerkin method, namely RDG (P1P2), is presented for time-accurate solutions to the compressible Navier-Stokes equations on 3D hybrid grids. The spatial discretization is carried out using a Taylor-basis discontinuous Galerkin method, in which a quadratic polynomial (P2) solution is reconstructed via a WENO (P1P2) reconstruction scheme from the underlying linear polynomial discontinuous Gakerkin (P1) solution in each cell for evaluating the fluxes. A series of Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta schemes (termed ESDIRK) are applied for temporal discretization to the resulting ordinary differential equations. The resulting non-linear system of equations at each stage is solved using an approximate Newton’s method, in which an LU-SGS preconditioned GMRES solver is applied for the solution to the linear system of equations. The developed code is applied to compute a series of benchmark test cases, including the implicit large eddy simulation of a turbulent lid driven cavity. The numerical results indicate that the use of ESDIRK schemes leads to remarkable improvement in solution efficiency and temporal accuracy over its explicit counterpart. In addition, this implicit RDG (P1P2) method requires much less storage and computing time than the implicit DG (P2) method, resulting in a fast, third-order implicit discontinuous Galerkin method for computing unsteady flow problems.

A Class of Reconstructed Discontinuous Galerkin Methods for Compressible Flows on Arbitrary Grids

A class of reconstructed discontinuous Galerkin methods is described for solving compressible flow problems on arbitrary grids. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via a so-called in-cell reconstruction process. The devised in-cell reconstruction is aimed to augment the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. These three reconstructed discontinuous Galerkin methods are used to compute a variety of compressible flow problems on arbitrary meshes to assess their accuracy. The numerical experiments demonstrate that all three reconstructed discontinuous Galerkin methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy, efficiency, and robustness I. Abstract The discontinuous Galerkin methods 1-25 (DGM) have recently become popular for the solution of systems of conservation laws. Nowadays, they are widely used in computational fluid dynamics, computational acoustics, and computational electromagnetics. The discontinuous Galerkin methods combine two advantageous features commonly associated to finite element and finite volume methods. As in classical finite element methods, accuracy is obtained by means of high-order polynomial approximation within an element rather than by wide stencils as in the case of finite volume methods. The physics of wave propagation is, however, accounted for by solving the Riemann problems that arise from the discontinuous representation of the solution at element interfaces. In this respect, the methods are therefore similar to finite volume methods. The discontinuous Galerkin methods have many attractive features:1) They have several useful mathematical properties with respect to conservation, stability, and convergence; 2) The method can be easily extended to higher-order (>2 nd

OpenACC-based GPU Acceleration of a p-multigrid Discontinuous Galerkin Method for Compressible Flows on 3D Unstructured Grids

A GPU accelerated p-multigrid discontinuous Galerkin (DG) method based on the OpenACC directives is presented for compressible flows on 3-D unstructured grids. The present design is aimed to utilize the power of high-performance GPU computing with very little intrusion and algorithm alteration to a well-developed CPU-based code. Due to the fact that the GPU memory is still far from abundant for high-order DG methods even on a top-rank model, a p-multigrid technique is therefore preferred for convergence acceleration rather than an implicit algorithm that requires huge memory for storing high-order Jacobian matrices. In this study, a multi-stage explicit time stepping scheme is used for advancing the higher-order approximation in time, with a first-order matrix-free implicit backward Euler scheme applied to accelerate the lower-order approximation. A variety of inviscid flow problems are computed on an NVIDIA Tesla K20c GPU to assess the performance of the developed GPU-accelerated code using a strong scaling test. The numerical results indicate that the p-multigrid discontinuous Galerkin method can be effectively accelerated on GPU in comparison with two eight-core AMD Opteron-6128 CPUs for its CPU-based counterpart.

On the Multi-GPU Computing of a Reconstructed Discontinuous Galerkin Method for Compressible Flows on 3D Hybrid Grids

A multi-GPU accelerated, third-order, reconstructed discontinuous Galerkin method, namely RDG(P1P2), has been developed based on the OpenACC directives for compressible flows on 3D hybrid grids. The present scheme requires minimum intrusion and algorithm alteration to an existing CPU code, which renders an efficient design approach for upgrading a legacy CFD solver with the GPU-computing capability while maintaining its portability across multiple platforms. The grid partitioning is performed according to the number of GPUs, and loaded equally on each GPU. Communication between the GPUs is achieved via the host-based MPI. A face renumbering and grouping algorithm is used to eliminate memory contention due to vectorized computing over the face loops on each individual GPU. A series of inviscid and viscous flow problems have been presented for the verification and scaling test, demonstrating excellent scalability of the resulting GPU code. The numerical results indicate that this parallel RDG(P1P2) method is a cost-effective, high-order DG method for scalable computing on GPU clusters.

A Hierarchical WENO Reconstructed Discontinuous Galerkin Method for Computing Shock Waves

The discontinuous Galerkin (DG) methods[1] have recently become popular for the solution of systems of conservation laws because of their several attractive features such as easy extension to and compact stencil for higher-order (> 2nd) approximation, flexibility in handling arbitrary types of grids for complex geometries, and amenability to parallelization and hp-adaptation. However, the DG Methods have their own share weaknesses. In particular, how to effectively control spurious oscillations in the presence of strong discontinuities, and how to reduce the computing costs and storage requirements for the DGM remain the two most challenging and unresolved issues in the DGM.

A Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Unstructured Tetrahedral Grids

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on unstructured tetrahedral grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on unstructured grids. The preliminary results indicate that this RDG method is stable on unstructured tetrahedral grids, and provides a viable and attractive alternative for the discretization of the viscous and heat fluxes in the Navier-Stokes equations.