High-Order DNS and LES Simulations Using an Implicit Tensor-Product Discontinuous Galerkin Method

Pazner, Will; Persson, Per‐Olof

doi:10.2514/6.2017-3948

Cited by 4 publications

(5 citation statements)

References 28 publications

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“…Significant research has also been undertaken into high-order methods with implicit time-stepping [24,25,26,27,28,29]. However, published research on their implementation for modern hardware appears to be very limited because of challenges related to very high storage requirements and increased global communication.…”

Section: Modern Hardwarementioning

confidence: 99%

High-Order Incompressible Computational Fluid Dynamics on Modern Hardware Architectures

Loppi¹

2021

Preprint

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Modern GPU architectures are characterised by an abundance of compute capability relative to memory bandwidth. This makes them very well-suited to solving temporally explicit and spatially compact discretisations of hyperbolic conservation laws. However, classical pressure-projection-based incompressible Navier–Stokes formulations do not fall into this category. One attractive formulation for solving incompressible problems on modern hardware is the method of artificial compressibility. When combined with explicit dual time stepping and a high-order Flux Reconstruction discretisation, the majority of operations can be cast as arithmetically intensive matrix–matrix multiplications that are well-suited for GPUs. In this seminar, I will present the high-order cross-platform incompressible Navier–Stokes solver in PyFR, together with three explicit convergence acceleration techniques: a polynomial multigrid, a novel locally adaptive pseudo-time stepping approach and novel stability-optimised Runge-Kutta schemes. The solver and the convergence acceleration techniques are validated for a range of turbulent test cases, including a simulation of the DARPA SUBOFF submarine model using hundreds of NVIDIA GPUs.

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Section: Modern Hardwarementioning

confidence: 99%

High-Order Incompressible Computational Fluid Dynamics on Modern Hardware Architectures

Loppi¹

2021

Preprint

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“…For a final set of test cases, we consider the compressible, inviscid Taylor-Green vortex (TGV) [39] at different Mach numbers. This problem has been extensively studied for the incompressible case [2], as well as the nearly-incompressible case [35,6,25,27]. The stability of DG discretizations for the under-resolved simulation of the inviscid TGV has also been studied in [43,21].…”

Section: D Isentropic Vortexmentioning

confidence: 99%

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

Pazner

Persson

2019

J Sci Comput

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We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropystable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions. arXiv:1809.09815v2 [math.NA]

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“…Boundary conditions are enforced by appropriate modification of these numerical flux functions. By setting τ h = ∇v h in equation (22) and inserting the resulting expression into (21), and subsequently integrating by parts again, we can eliminate σ h to obtain the primal formulation…”

Section: Iia Treatment Of Second-order Termsmentioning

confidence: 99%

“…25 A challenge often encountered when using matrix-free methods such as the method described above is the construction of a preconditioner without having access to the entries of the matrix. In our previous works, 20,21 we developed a strategy for approximating the diagonal blocks of the DG Jacobian matrix without needing to explicitly form the matrix.…”

Section: Tensor-product Preconditionersmentioning

confidence: 99%

“…In our previous works, 20,21 we introduced a new Kronecker-product based preconditioner that exploits the tensor-product structure of nodal discontinuous Galerkin discretizations on quadrilateral or hexahedral meshes. This preconditioner generates optimal tensor-product approximations to the diagonal blocks of the Jacobian matrix, with the advantage that it greatly reduces the computational complexity associated with forming and applying it when compared with the exact block Jacobi method.…”

Section: Introductionmentioning

confidence: 99%

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Interior penalty tensor-product preconditioners for high-order discontinuous Galerkin discretizations

Pazner

Persson

2018

2018 AIAA Aerospace Sciences Meeting

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In this paper, we introduce an interior penalty tensor-product preconditioner for the implicit time integration of discontinuous Galerkin discretizations of partial differential equations with second-order spatial derivatives. This preconditioner can be efficiently formed using a sum-factorized Lanczos algorithm for computing the Kronecker-product singular value decomposition of the diagonal blocks of the Jacobian matrix, and can be applied efficiently using a simultaneous triangularization procedure. In two spatial dimensions, the computational complexity for the overall method is linear per degree of freedom, which is the same as that of a sum-factorized explicit method. This preconditioner exactly reproduces the block Jacobi preconditioner for certain special cases, and compares favorably with the block Jacobi preconditioner for a range of test problems, including viscous compressible flow over a circular cylinder. This preconditioner shows greatly improved performance when compared with a Kronecker-product preconditioner that only incorporates first-order terms.

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High-Order DNS and LES Simulations Using an Implicit Tensor-Product Discontinuous Galerkin Method

Cited by 4 publications

References 28 publications

High-Order Incompressible Computational Fluid Dynamics on Modern Hardware Architectures

High-Order Incompressible Computational Fluid Dynamics on Modern Hardware Architectures

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

Interior penalty tensor-product preconditioners for high-order discontinuous Galerkin discretizations

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