Fourth-order runge-kutta schemes for fluid mechanics applications

Carpenter, Mark H.; Kennedy, Christopher; Bijl, H.; Viken, Sally A.; Vatsa, Veer N.

doi:10.1007/bf02728987

Cited by 30 publications

(54 citation statements)

References 24 publications

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“…Afterwards, the efficiency of the proposed DG-Peer coupling was investigated for a turbulent test case by means of the adaptive time-step strategy and compared with the traditional one-step temporal schemes, i.e. the fourth order/six stages ESDIRK scheme [13] and the third order/three stages (ROS3PL) [23] and fourth order/six stages (RODASP) [30] linearly implicit one-step Rosenbrock schemes.…”

Section: Resultsmentioning

confidence: 99%

“…Also the circular cylinder test case has been used to investigate the computational efficiency of Peer methods with the adaptive time-step strategy. A comparative assessment in terms of accuracy and performance with the traditional fourth order/six stages ESDIRK scheme [13] and third order/three stages (ROS3PL) [23] and fourth order/six stages (RODASP) [30] linearly implicit one-step Rosenbrock schemes has been done.…”

Section: Introductionmentioning

confidence: 99%

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High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Massa¹,

Noventa²,

Bassi³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. In recent years the increasing attention to high-order Finite Volume (FV), Finite Element (FE) and spectral methods and the growth of computing power promote the development of high-order temporal schemes to perform robust, accurate and efficient unsteady long-time simulations. In this context, some features of the Discontinuous Galerkin finite element (DG) methods, e.g. compactness and flexibility, can be advantageous both for explicit and implicit time integration approaches. Explicit schemes can achieve very high accuracy, but are limited by time-step restrictions, while implicit schemes, even if memory consuming, can overcome time-step limitations, thus improving the time integration efficiency. During last decades several high order implicit temporal schemes have been developed, and some of them have been successfully coupled with DG methods. However these schemes can show the order reduction if applied to very stiff problems or problems with time-dependent boundary conditions. To overcome these limitations, high-order linearly implicit two-step peer methods have been proposed and successfully applied to the numerical solution of differential-algebraic equations. The aim of the present work is to implement high-order two-step peer methods in a DG code and assess their performance for the unsteady solution of the incompressible Navier-Stokes (INS) and Reynolds Averaged Navier-Stokes equations (RANS) equations.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Massa¹,

Noventa²,

Bassi³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…A family of implicit Runge-Kutta schemes with L-stability has been recently developed by Carpenter and coauthors (5,6) . These "explicit, singly diagonal implicit Runge-Kutta" (ESDIRK) schemes can be written as (13) where s is the number of stages; and , , are the stage, the main, and the embedded scheme weights, respectively.…”

Section: Implicit Runge-kutta (Esdirk) Methodsmentioning

confidence: 99%

“…It can be shown that the recovered in-cell distribution is (R+1) th -order-accurate, where . The recovery DoFs are computed using the following "weak statement": (6) and (7) The first N DoFs of the unlimited rDG coincide with DG's DoFs, (8) The rest of the DoFs are given in Appendix A for .…”

Section: B "Recovery" Discontinuous Galerkin (Rdg) Familymentioning

confidence: 99%

Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

Theofanous

Park

Mousseau

et al. 2008

46th AIAA Aerospace Sciences Meeting and Exhibit

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In recent work (Nourgaliev, Liou, Theofanous, JCP in press) we demonstrated that numerical simulations of interfacial flows in the presence of strong shear must be cast in dynamically sharp terms (sharp interface treatment or SIM), and that moreover they must meet stringent resolution requirements (i.e., resolving the critical layer). The present work is an outgrowth of that work aiming to overcome consequent limitations on the temporal treatment, which become still more severe in the presence of phase change. The key is to avoid operator splitting between interface motion, fluid convection, viscous/heat diffusion and reactions; instead treating all these non-linear operators fully-coupled within a Newton iteration scheme. To this end, the SIM's cut-cell meshing is combined with the high-orderaccurate implicit Runge-Kutta and the "recovery" Discontinuous Galerkin methods along with a Jacobian-free, Krylov subspace iteration algorithm and its physics-based preconditioning. In particular, the interfacial geometry (i.e., marker's positions and volumes of cut cells) is a part of the Newton-Krylov solution vector, so that the interface dynamics and fluid motions are fully-(non-linearly)-coupled. We show that our method is: (a) robust (L-stable) and efficient, allowing to step over stability time steps at will while maintaining high-(up to the 5 th )-order temporal accuracy; (b) fully conservative, even near multimaterial contacts, without any adverse consequences (pressure/velocity oscillations); and (c) highorder-accurate in spatial discretization (demonstrated here up to the 12 th -order for smoothin-the-bulk-fluid flows), capturing interfacial jumps sharply, within one cell. Performance is illustrated with a variety of test problems, including low-Mach-number "manufactured" solutions, shock dynamics/tracking with slow dynamic time scales, and multi-fluid, highspeed shock-tube problems. We briefly discuss preconditioning, and we introduce two physics-based preconditioners -"Block-Diagonal" and "Internal energy-Pressure-Velocity Partially Decoupled", demonstrating the ability to efficiently solve all-speed flows with strong effects from viscous dissipation and heat conduction.

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“…Representative classes of time-integration schemes embedded in the GL method consist of implicit multistep methods such as Adams-Moulton (AM) [22] and backward differentiation (BDF) methods [13,20,21], implicit multistage Runge-Kutta schemes such as diagonally (DIRK) and singly-diagonally (SDIRK) implicit Runge-Kutta schemes [3,19,59], explicit multistep methods, such as leapfrog and Adams-Bashforth methods [28,43], explicit Runge-Kutta schemes, such as the fourthorder Runge-Kutta scheme [55] and partitioned methods, such as Implicit-Explicit (IMEX) schemes, whereby the operators are linearized in some fashion with-e.g., two Butcher tableaux, one explicit and one implicit [5,40,106]. While EBTI schemes are widely used in computational fluid dynamics, especially in the engineering sector [18,52], their adoption in the weather and climate communities has been less widespread, with SE schemes [54,88,107] and horizontally-explicit vertically-implicit schemes [8,40,63]-i.e., schemes where the horizontal direction is treated explicitly and the vertical is treated implicitly-becoming more prominent but still confined mainly to research and limited-area models (with very few exceptions-see Table 1). Within this context, Eq.…”

Section: Eulerian-based Time-integration (Ebti)mentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

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The continuous partial differential equations governing a given physical phenomenon, such as the Navier-Stokes equations describing the fluid motion, must be numerically discretized in space and time in order to obtain a solution otherwise not readily available in closed (i.e., analytic) form. While the overall numerical discretization plays an essential role in the algorithmic efficiency and physically-faithful representation of the solution, the time-integration strategy commonly is one of the main drivers in terms of cost-to-solution (e.g., time-or energy-to-solution), accuracy and numerical stability, thus constituting one of the key building blocks of the computational model. This is especially true in time-critical applications, including numerical weather prediction (NWP), climate simulations and engineering. This review provides a comprehensive overview of the existing and emerging time-integration (also referred to as time-stepping) practices used in the operational global NWP and climate industry, where global refers to weather and climate simulations performed on the entire globe. While there are many flavors of time-integration strategies, in this review we focus on the most widely adopted in NWP and climate centers and we emphasize the reasons why such numerical solutions were adopted. This allows us to make some considerations on future trends in the field such as the need to balance accuracy in time with substantially enhanced time-to-solution and associated implications on energy consumption and running costs. In addition, the potential for the co-design of time-stepping algorithms and underlying high performance computing hardware, a keystone to accelerate the computational performance of future NWP and climate services, is also discussed in the context of the demanding operational requirements of the weather and climate industry.

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Fourth-order runge-kutta schemes for fluid mechanics applications

Cited by 30 publications

References 24 publications

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

High-Order Linearly Implicit Two-Step Peer Methods for the Discontinuous Galerkin Solution of the Incompressible Rans Equations

Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

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