Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion, Michael L.; Saye, Robert I.

doi:10.1016/j.jcp.2018.08.054

Cited by 18 publications

(31 citation statements)

References 68 publications

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“…( 29) is integrated in time using high-order TVD Runge-Kutta schemes, which are very efficient for many practical applications but are limited to fourth order accuracy. Methods such as the spectral deferred correction methods [76,77,78] or ADER schemes [79,80,74,75] could be implemented within the present framework to obtain space-time orders of accuracy that are higher than the ones considered here.…”

Section: Discussionmentioning

confidence: 99%

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Gulizzi,

Almgren,

Bell

2021

Preprint

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We present a computational framework for solving the equations of inviscid gas dynamics using structured grids with embedded geometries. The novelty of the proposed approach is the use of high-order discontinuous Galerkin (dG) schemes and a shock-capturing Finite Volume (FV) scheme coupled via an hp adaptive mesh refinement (hp-AMR) strategy that offers high-order accurate resolution of the embedded geometries. The hp-AMR strategy is based on a multi-level block-structured domain partition in which each level is represented by block-structured Cartesian grids and the embedded geometry is represented implicitly by a level set function. The intersection of the embedded geometry with the grids produces the implicitly-defined mesh that consists of a collection of regular rectangular cells plus a relatively small number of irregular curved elements in the vicinity of the embedded boundaries. High-order quadrature rules for implicitly-defined domains enable high-order accuracy resolution of the curved elements with a cell-merging strategy to address the small-cell problem. The hp-AMR algorithm treats the system with a second-order finite volume scheme at the finest level to dynamically track the evolution of solution discontinuities while using dG schemes at coarser levels to provide high-order accuracy in smooth regions of the flow. On the dG levels, the methodology supports different orders of basis functions on different levels. The space-discretized governing equations are then advanced explicitly in time using high-order Runge-Kutta algorithms. Numerical tests are presented for two-dimensional and three-dimensional problems involving an ideal gas. The results are compared with both analytical solutions and experimental observations and demonstrate that the framework provides high-order accuracy for smooth flows and accurately captures solution discontinuities.

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Section: Discussionmentioning

confidence: 99%

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Gulizzi,

Almgren,

Bell

2021

Preprint

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“…Each sweep solves Eq. ( 10) for m = 1 to M and increases the order by one [13], although stiffness or time-dependent boundary conditions can degrade convergence [8,35]. The converged solution satisfies the Lobatto IIIA collocation scheme with M stages and, hence, achieves order 2M at the end of the time interval [20].…”

Section: Spectral Deferred Correctionmentioning

confidence: 96%

“…The first problem represents a 2D Taylor-Green vortex that travels with a phase speed of one in the x and y directions. It was proposed by Minion and Saye [35] and possesses the exact solution…”

Section: Traveling Taylor-green Vortexmentioning

confidence: 99%

“…is decomposed into 8 × 8 × 1 cubic elements of degree P = 10. Following [35] two test cases are defined: TGP with all directions periodic and kinematic viscosity ν = 0.02, and TGD with Dirichlet conditions in y direction, x, z periodic and ν = 0.01. Both cases are initialized with the exact solution and integrated with steps ranging from ∆t = 2 −5 to 2 −13 until reaching the final time of T = 0.25.…”

Section: Traveling Taylor-green Vortexmentioning

confidence: 99%

“…This method has excellent stability properties that are comparable with IMEX RK methods, but extends easily to higher order. It was adopted for incompressible flows in periodic domains in [37] and later extended to time-dependent boundary conditions [35]. In [53] the SISDC method was generalized to variable properties and demonstrated to reach convergence rates up to order 12 for three-dimensional vortical flows with nonlinear (turbulent) viscosity.…”

Section: Introductionmentioning

confidence: 99%

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Assessment of high-order IMEX methods for incompressible flow

Montadhar¹,

Grotteschi²,

Stiller³

2021

Preprint

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This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight-forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time-dependent boundary conditions, two thirdorder RK methods are identified that perform well in all test cases and clearly surpass all second-order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca 10 −5 .

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Assessment of high‐order IMEX methods for incompressible flow

Montadhar

Grotteschi

Stiller

2023

Numerical Methods in Fluids

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This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight-forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time-dependent boundary conditions, two third-order RK methods are identified that perform well in all test cases and clearly surpass all second-order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca 10 −5 .

show abstract

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Cited by 18 publications

References 68 publications

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Assessment of high-order IMEX methods for incompressible flow

Assessment of high‐order IMEX methods for incompressible flow

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