Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

Kubatko, Ethan J.; Yeager, Benjamin; Ketcheson, David I.

doi:10.1007/s10915-013-9796-7

Cited by 52 publications

(32 citation statements)

References 35 publications

(55 reference statements)

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“…In this work, we use the low‐storage/balanced schemes derived in , in particular the third‐order, three‐stage scheme presented in Section 3.2 of and method RK4(3)5[2R+] of order four with five stages. As an alternative, the strong‐stability preserving Runge–Kutta schemes particularly derived for transport problems with DG from with a variable number of stages to a fixed order of accuracy (e.g., q = 8 for order 4) were also tested. The additional stages are used to extend the domain of linear stability, which can improve overall efficiency.…”

Section: Explicit and Implicit Runge–kutta Time Steppingmentioning

confidence: 99%

Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation

Kronbichler

Schoeder

Müller

et al. 2015

Numerical Meth Engineering

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Summary We describe implicit and explicit formulations of the hybridizable discontinuous Galerkin method for the acoustic wave equation based on state‐of‐the‐art numerical software and quantify their efficiency for realistic application settings. In the explicit scheme, the trace of the acoustic pressure is computed from the solution on the two elements adjacent to the face at the old time step. Tensor product shape functions for quadrilaterals and hexahedra evaluated with sum factorization are used to ensure low operation counts. For applying the inverse mass matrix of Lagrangian shape functions with full Gaussian quadrature, a new tensorial technique is proposed. As time propagators, diagonally implicit and explicit Runge–Kutta methods are used, respectively. We find that the computing time per time step is 25 to 200 times lower for the explicit scheme, with an increasing gap in three spatial dimensions and for higher element degrees. Our experiments on realistic 3D wave propagation with variable material parameters in a photoacoustic imaging setting show an improvement of two orders of magnitude in terms of time to solution, despite stability restrictions on the time step of the explicit scheme. Operation counts and a performance model to predict performance on other computer systems accompany our results. Copyright © 2015 John Wiley & Sons, Ltd.

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Section: Explicit and Implicit Runge–kutta Time Steppingmentioning

confidence: 99%

Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation

Kronbichler

Schoeder

Müller

et al. 2015

Numerical Meth Engineering

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“…Using an explicit, s ‐stage Runge‐Kutta time integration scheme in Butcher notation, the time discrete problem is given as

\begin{matrix} right {\underset{true_}{k}}^{(i)} & left = \underset{true_}{L} ({\underset{true_}{U}}_{n} + Δ t \sum_{j = 1}^{i - 1} a_{i j} {\underset{true_}{k}}^{(j)}, t_{n} + c_{i} Δ t), i = 1, ⋯, s, & right \\ right {\underset{true_}{U}}_{n + 1} & left = {\underset{true_}{U}}_{n} + Δ t \sum_{i = 1}^{s} b_{i} {\underset{true_}{k}}^{(i)}, \end{matrix}

with

c_{i} = \sum_{j = 1}^{s} a_{i j}

and a i j = 0 for j ≥ i . We implemented and tested Runge‐Kutta methods developed in the works of Kennedy et al, Toulorge and Desmet, and Kubatko et al While the methods developed in the work of Kennedy et al are generic in the sense that they are not designed for a specific class of discretization methods, the methods proposed in the works of Toulorge and Desmet and Kubatko et al aim at maximizing the critical time step size in view of DG discretization methods. Furthermore, the low‐storage Runge‐Kutta methods developed in the work of Toulorge and Desmet have the advantage that they involve less vector update operations per time step as compared to the strong‐stability–preserving Runge‐Kutta methods in the work of Kubatko et al For this reason and in agreement with our numerical experiments, the low‐storage explicit Runge‐Kutta methods of Toulorge and Desmet appear to be the most efficient time integration schemes.…”

Section: Compressible Navier‐stokes Dg Solvermentioning

confidence: 99%

“…We implemented and tested Runge‐Kutta methods developed in the works of Kennedy et al, Toulorge and Desmet, and Kubatko et al While the methods developed in the work of Kennedy et al are generic in the sense that they are not designed for a specific class of discretization methods, the methods proposed in the works of Toulorge and Desmet and Kubatko et al aim at maximizing the critical time step size in view of DG discretization methods. Furthermore, the low‐storage Runge‐Kutta methods developed in the work of Toulorge and Desmet have the advantage that they involve less vector update operations per time step as compared to the strong‐stability–preserving Runge‐Kutta methods in the work of Kubatko et al For this reason and in agreement with our numerical experiments, the low‐storage explicit Runge‐Kutta methods of Toulorge and Desmet appear to be the most efficient time integration schemes. Hence, we use the third‐order p = 3 low‐storage explicit Runge‐Kutta method with s = 7 stages of Toulorge and Desmet that resulted in the best performance in our preliminary tests, ie, it allows time step sizes as large as for strong‐stability–preserving Runge‐Kutta methods but requires less memory and less vector update operations.…”

Section: Compressible Navier‐stokes Dg Solvermentioning

confidence: 99%

“…We consider explicit Runge-Kutta time integration methods for discretization in time. [26][27][28] The explicit treatment of convective terms and viscous terms sets an upper bound on the time step size in order to maintain stability. As discussed below and thoroughly investigated in this work, the time step limitation related to the viscous term is less restrictive than the convective one for flow problems characterized by low viscosities and coarse spatial resolutions.…”

Section: Temporal Discretizationmentioning

confidence: 99%

“…For the Re = 1 test case, we reduce the end time of the simulation to t f = t 0 in order to reduce costs. On the basis of our experience, we assume exponents of e = 1.5 and f = 3.0 in Equation (28) and verify this assumption numerically.…”

Section: Selection Of Time Step Sizementioning

confidence: 99%

See 2 more Smart Citations

A matrix‐free high‐order discontinuous Galerkin compressible Navier‐Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Fehn

Wall

Kronbichler

2018

Numerical Methods in Fluids

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Summary Both compressible and incompressible Navier‐Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high‐order discontinuous Galerkin (DG) discretizations of the compressible Navier‐Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high‐performance DG solver for the compressible Navier‐Stokes equations based on a highly efficient matrix‐free implementation that targets modern cache‐based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node‐level performance, and our results suggest that there is great potential for further performance improvements for current state‐of‐the‐art DG implementations of the compressible Navier‐Stokes equations. Secondly, this compressible Navier‐Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix‐free implementation. We discuss algorithmic differences between both solution strategies and present an in‐depth numerical investigation of the performance. The considered benchmark test cases are the three‐dimensional Taylor‐Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall‐bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.

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The Discontinuous Galerkin Method: Derivation and Properties

Kronbichler

2021

Efficient High-Order Discretizations for Computational Fluid Dynamics

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Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

Cited by 52 publications

References 35 publications

Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation

Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation

A matrix‐free high‐order discontinuous Galerkin compressible Navier‐Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

The Discontinuous Galerkin Method: Derivation and Properties

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