Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations

doi:10.1016/S0168-9274(99)00141-5

Applied Numerical Mathematics

Volume 35, Issue 3, November 2000, Pages 177-219

https://doi.org/10.1016/S0168-9274(99)00141-5 Get rights and content

Abstract

The derivation of low-storage, explicit Runge–Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier–Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, sixteen ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been tested with not only DETEST, but also with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be nearly matched while still saving 2–3 registers of memory.

References (95)

I. Alonso-Mallo
Explicit single-step methods with optimal order of convergence for partial differential equations
Appl. Numer. Math.
(1999)
T.S. Baker et al.
Continuous approximation with embedded Runge–Kutta methods
Appl. Numer. Math.
(1996)
P. Bogacki et al.
A (3,2) pair of Runge–Kutta formulas
Appl. Math. Lett.
(1989)
P. Bogacki et al.
An efficient Runge–Kutta (4,5) pair
Comput. Math. Appl.
(1996)
M. Calvo et al.
Explicit Runge–Kutta methods for initial problems with oscillating solutions
J. Comput. Appl. Math.
(1996)
J.R. Dormand et al.
A family of embedded Runge–Kutta formulae
J. Comput. Appl. Math.
(1980)
J.R. Dormand et al.
A reconsideration of some embedded Runge–Kutta formulae
J. Comput. Appl. Math.
(1986)
I.R. Gran
Negative flame speed in an unsteady 2-D premixed flame: A computational study
D.J. Higham
Regular Runge–Kutta pairs
Appl. Numer. Math.
(1997)
Z. Horváth
Positivity of Runge–Kutta and diagonally split Runge–Kutta methods
Appl. Numer. Math.
(1998)

M.E. Hosea et al.

Efficiency comparisons of methods for integrating ODEs

Comput. Math. Appl.

(1994)

F.Q. Hu et al.

Low-dissipation and low-dispersion Runge–Kutta schemes for computational acoustics

J. Comput. Phys.

(1996)

C.A. Kennedy et al.

A comparison of several new numerical methods for the simulation of compressible shear layers

Appl. Numer. Math.

(1994)

J.F.B.M. Kraaijevanger et al.

Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations

J. Comput. Appl. Math.

(1987)

P.J. Prince et al.

High order embedded Runge–Kutta formulae

J. Comput. Appl. Math.

(1981)

C.-W. Shu et al.

Efficient implementation of essentially non-oscillatory schemes

J. Comput. Phys.

(1988)

D. Stanescu et al.

2N-storage low dissipation and dispersion Runge–Kutta schemes for computational acoustics

J. Comput. Phys.

(1998)

F. Vadillo

On spurious fixed points of Runge–Kutta methods

J. Comput. Phys.

(1997)

J.H. Verner

High-order explicit Runge–Kutta pairs with low stage order

Appl. Numer. Math.

(1996)

J.H. Williamson

Low-storage Runge–Kutta schemes

J. Comput. Phys.

(1980)

M. Zennaro

Contractivity of Runge–Kutta methods with respect to forcing terms

Appl. Numer. Math.

(1993)

D.W. Zingg et al.

Runge–Kutta methods for linear ordinary differential equations

Appl. Numer. Math.

(1999)

S. Abarbanel et al.

On the removal of boundary errors caused by Runge–Kutta integration of nonlinear partial differential equations

SIAM J. Sci. Comput.

(1996)

M.A. Aves et al.

Does error control suppress spuriosity?

SIAM J. Numer. Anal.

(1997)

G.A. Blaisdell et al.

Compressibility effects on the growth and structure of homogeneous turbulent shear flow

J. Fluid Mech.

(1993)

J.C. Butcher

Coefficients for the study of Runge–Kutta integration processes

J. Austral. Math. Soc. Ser. B

(1964)

J.C. Butcher

On Runge–Kutta processes of high order

J. Austral. Math. Soc. Ser. B

(1964)

J.C. Butcher

The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods

(1987)

M.H. Carpenter et al.

Fourth-order 2N-storage Runge–Kutta schemes, NASA TM-109112

(1994)

M.H. Carpenter et al.

Third-order 2N-storage Runge–Kutta schemes with error control, NASA TM-109111

(1994)

M.H. Carpenter et al.

The theoretical accuracy of Runge–Kutta time discretizations for the initial boundary value problem: a study of the boundary error

SIAM J. Sci. Comput.

(1995)

S.D. Conte et al.

A Kutta third-order procedure for solving differential equations requiring minimum storage

SIAM J. Numer. Anal.

(1956)

G.J. Cooper

A generalization of algebraic stability for Runge–Kutta methods

IMA J. Numer. Anal.

(1984)

G.G. Dahlquist et al.

Generalized disks of contractivity for explicit and implicit Runge–Kutta methods, Report TRITA-NA-7906

(1979)

K. Dekker et al.

Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations

(1984)

J.E. Dennis et al.

Numerical Methods for Unconstrained Optimization and Nonlinear Equations

(1983)

J.R. Dormand

Numerical Methods for Differential Equations

(1996)

W.H. Enright et al.

Interpolants for Runge–Kutta formulas

ACM Trans. Math. Software

(1986)

W.H. Enright et al.

Two FORTRAN packages for assessing initial value methods

ACM Trans. Math. Software

(1987)

W.H. Enright et al.

A survey of the explicit Runge–Kutta method, Technical Report, 291/94

(1994)

G. Erlebacher et al.

Interaction of a shock with a longitudinal vortex

J. Fluid Mech.

(1997)

E. Fehlberg

Low-order classical Runge–Kutta formulas with stepsize control and their application to some heat transfer problems, NASA Technical Report TR R-315

(1969)

D.J. Fyfe

Economical evaluation of Runge–Kutta formulae

Math. Comp.

(1966)

D.M. Gay

Computing optimal locally constrained steps

SIAM J. Sci. Statist. Comput.

(1981)

D.M. Gay

ALGORITHM 611. Subroutines for unconstrained minimization using a model/trust region approach

ACM Trans. Math. Software

(1983)

S. Gill

A process for the step-by-step integration of differential equations in an automatic digital computing machine

Proc. Cambridge Phil. Soc.

(1951)

S. Gottlieb et al.

Total variation diminishing Runge–Kutta schemes

Math. Comp.

(1998)

Cited by (498)

RANSBox: A zero-dimensional modular software package for Reynolds-averaged Navier-Stokes modeling
2024, Computer Physics Communications
RANSBox is a zero-dimensional software package for Reynolds-averaged Navier-Stokes (RANS) modeling developed at Lawrence Livermore National Laboratory to support common implementation of RANS models across a variety of host codes with different numerical schemes and code bases. This work describes the key features of RANSBox including “model-agnostic integration,” which allows new models to be implemented in RANSBox and quickly deployed to host codes without additional changes to the host code base. Three one-dimensional test problems with analytical asymptotic properties are described which can be used to verify correctness of RANSBox integration. Results for these problems are then compared across different RANS models in a single code and across different codes with a common RANS model. By using a common model implementation in RANSBox, host codes with different numerical schemes and formal orders of accuracy are demonstrated to predict the expected behavior for the three test problems.
The effect of basis polynomial degree on the performance of compressible flow simulations employing a split-form DG method
2024, Computers and Fluids
High order methods have garnered significant interest recently for simulating compressible flows due to their low dissipation nature. In this regard, the use of discontinuous Galerkin method has been actively pursued due to the advantages ensuing from its compact interpolation. However, these methods generally require employing a limiter for stable computation of supersonic compressible flows that contain shocks. Many limiters are designed to incorporate ideas from robust second order implementations of the Finite Volume Method for shock capturing. It is therefore prudent to examine the effectiveness of high order discontinuous Galerkin solutions when simulating flows containing discontinuities. To this end, the present work investigates the variation of accuracy per degrees of freedom as the basis polynomial degree is changed. Several standard one-dimensional and two-dimensional compressible flow cases are simulated for multiple basis polynomial degrees while employing different meshes to keep the degrees of freedom fixed. It is found that high order solutions are more efficient than second order solution as regards the accuracy per degrees of freedom. The improvement in this metric with increasing basis polynomial degree plateaus at cubic basis degree for cases containing strong discontinuities (shock Mach numbers greater than 3), and shifts to a further higher basis in simpler problems. However, in problems containing slowly moving shocks and or weak discontinuities, high order solutions generate increased numerical oscillations, suggesting the need for improvement in anti-aliasing techniques.
Combustion mechanism of ammonia and ammonia-hydrogen mixtures in the intensely turbulent jet mixing layer under the spontaneous ignition condition
2024, International Journal of Hydrogen Energy
The higher-order DNS simulations were carried out to study the turbulent jet combustion characteristics of pure ammonia and ammonia-hydrogen mixtures at the auto-ignition condition. It is found that blending just 5% H₂ into ammonia can significantly shorten its ignition delay time, enhance the reactive scalar dissipation rate, and thus produce much more fine structures in the turbulence domain. During the turbulent ignition process, the low-temperature ignition firstly occurred at the most reactive mixture fraction (ξ_MR) that is located on the lean-side due to the spontaneous reactions, then the ignited reactive fronts propagated towards rich side by the deflagrative mode, which eventually ignited the high-temperature reactions near the stoichiometric mixture fraction (ξ_st). The advancement of early-phase ignition at ξ_MR as well as speed up of the subsequent deflagrative propagation was responsible for the acceleration of ammonia flame ignition in turbulence. The early-phase flame propagation was driven by chemical reactions, while in the high temperature post-ignition phase, flame curvature due to the sheared turbulence with a neutralizing balance with the normal diffusion was responsible for the motionlessness of reaction front at a fixed leaner mixture fraction near ξ_st. The majority triple points with moderate scalar dissipation rate (χ) were propagated by the normal edge-flame propagation mode; in the ultra-lean fluids with low χ, the edge-flame propagation was governed by expanding flame kernels; and in the fluids with high χ, flame folding played a leading role in the ξ_st iso-surface ignition. In overall, the flame folding regime was dominated in the turbulent edge flame propagation along the ξ_st iso-surface. In spite of its marginal contribution to the global heat production and temperature rise, the deflagrative mode plays a significant role in regulating the ignition timing and thus optimizing the ammonia engine performance.
A mesh-free framework for high-order direct numerical simulations of combustion in complex geometries
2024, Computer Methods in Applied Mechanics and Engineering
The multiscale nature of turbulent combustion necessitates accurate and computationally efficient methods for direct numerical simulations (DNS). The field has long been dominated by high-order finite differences, which lack the flexibility and adaptivity for simulations of complex geometries and flame-turbulence-structure interactions in realistic settings. In this work I introduce a new approach to DNS of premixed combustion, based on a high-order mesh-free discretisation in combination with finite differences, enabling high-order simulations in non-trivial geometries. The approach is validated against a range of two- and three-dimensional flows, both laminar and turbulent, and reacting and inert. The present method (a) has the resolving power for DNS of both laminar flames and inert turbulence with comparable accuracy to high-order finite differences, (b) can capture the dynamics of unsteady bluff body stabilised flames, and (c) is capable of simulating flame-turbulence interactions, with results comparing qualitatively well with published data. This work paves the way for DNS of combustion in complex geometries, offering an alternative approach to methods based on structured grids with immersed boundaries, or unstructured meshes. Further studies with the present method are proposed, which will aid understanding of fundamental flame dynamics in non-trivial geometries. Planned developments in adaptivity and extension of the mesh-free construction to all three dimensions will increase the value of the method, and support the push towards DNS of real geometries.
On the application of principal component transport for compression ignition of lean fuel/air mixtures under engine relevant conditions
2024, Combustion and Flame
Principal component transport-based data-driven reduced-order models (PC-transport ROM) are being increasingly adopted as a combustion model of turbulent reactive flows to mitigate the computational cost associated with incorporating detailed chemical kinetics. Previous studies were mainly limited to replicating relatively-simple chemistry in canonical configurations. The objective of the present study, therefore, is to further explore the accuracy of PC-transport ROM on more complex combustion phenomenon where, for example, large hydrocarbon fuel chemistry spanning a broad range of thermochemical space governs sequential multi-stage compression ignition processes. The cumulative error of PC-transport for this problem, and for others that depend upon sequential highly nonlinear physics, has to be minimal as the combustion phasing and heat release rate in internal combustion engines depends upon accurate predictions of minor ignition species whose concentrations start from ashes and grow orders of magnitude over the course of low- and high-temperature autoignition. Specifically, the PC-transport ROM is applied to predict the compression ignition characteristics of lean $n$ -heptane/air and primary reference fuel (PRF)/air mixtures in a two-dimensional (2-D) constant volume computational domain initialized with a two-dimensional isotropic turbulence spectrum and temperature inhomogeneities. PCA is used to define the low-dimensional manifold that represents the original thermochemical state vector, and artificial neural network (ANN) models are adopted to tabulate chemical kinetics, transport, and thermodynamic properties. A series of 2-D pseudo-turbulent simulations are performed at engine pressures by varying the initial mean and r.m.s. of temperature, turbulence intensity, and the composition of fuel/air mixture. The results show that the PC-transport ROM accurately reproduces the instantaneous and statistical ignition characteristics of the fuel/air mixture, aided by pre-processing techniques including species subsetting, data clustering, and data transformation. It is found that PCs are not properly scaled with a power transformer if reactants are included in the species subset, which leads to a decrease in the accuracy of the PC-transport ROM. A separation of the reactants from the species subset ensures that the temporal evolution of the PCs starts from zero and spans orders of magnitude with time, and as such, this approach is found to effectively redistribute both PCs and their source terms with a power transformer. The computational speed-up factor of the PC-transport ROM ranges between 5.1 and 15.0 for the cases with $n$ -heptane/air mixture and PRF/air mixture, respectively. Moreover, a potential further speed-up is anticipated through a combination of reduction in grid resolution requirements and in the stiffness of the chemical system. As an example, many of the pre-processing methods for inhomogeneous compression ignition may also apply to other complex intermittent combustion phenomena.
Novelty and significance statement
• The PCA-based reduced-order model (PC-transport ROM) has been applied to the multi-stage compression ignition of large hydrocarbon fuels under HCCI-relevant conditions. The present work presents a systematic procedure to accurately capture the two-stage ignition behavior of lean $n$ -heptane/air or PRF50/air mixture.
• The present work demonstrates an advantage of the PC-transport ROM in terms of computational speed-up. The computational speed-up factor for the ROM is up to 15, and moreover, a potential additional speed-up is anticipated through the reduction in the spatial and temporal resolution required.
• A series of 2-D PC-transport ROMs are conducted to demonstrate the robustness of the ROM. A limitation of the ROM against different operating conditions is also discussed.
libFastMesh: An optimized finite-volume framework for computational aeroacoustics
2024, Computer Physics Communications
This paper presents an optimized framework for simulating aeroacoustic flow on unstructured meshes. The libFastMesh (LFM) framework is based on a low-dissipation finite–volume discretization, together with high-order explicit time integration, for direct computation of aeroacoustic fields. These methods were previously implemented in caafoam, an OpenFOAM-based solver developed by D'Alessandro et al. [1] to resolve aeroacoustic flows. The new framework is specifically developed from scratch to alleviate two of the main bottlenecks currently limiting the performance of OpenFOAM-based solvers at large scale: memory bandwidth and inter-node communication. Compact data structures and compute kernel fusion are employed to enhance cache utilization and re-use, respectively. Separate treatment of inter-process boundary cells, together with non-blocking MPI communication enable effective overlap between communication and flux computations. The accuracy and robustness of libFastMesh are established via simulations of well-established aeroacoustic flow benchmarks. Comparison to previous simulation results obtained with caafoam, and to DNS data serve to validate the code. Finally, the effectiveness of the aforementioned optimizations is demonstrated through rigorous analysis of the proposed solver performance in single-node and multi-node operation modes. The obtained results show that libFastMesh offers a speed-up of up to 20x with respect to the previously developed OpenFOAM-based solver, caafoam, in large-scale computations. Moreover, LFM is shown to scale extremely for cell-per-core ratios of less than 500. These results are particularly appealing given the highly resource-demanding nature of direct computational aeroacoustics (CAA). Consequently, LFM could be an extremely attractive tool for scientists who wish to conduct large-scale CAA simulations on modern, exascale architectures.
Program Title: libFastMesh
CPC Library link to program files: https://doi.org/10.17632/7w9fy2xtcf.1
Developer's repository link: https://github.com/TRC-HPC/LFM_Public
Licensing provisions: GPLv3
Programming language: C++
Nature of problem: This software solves compressible Navier–Stokes equations. The adopted numerics and flow models allow to capture with high accuracy aerodinamically generated sound. The solver is also designed to take advantage of the massively parallel computing systems which are strictly required for facing real–life aeroacoustic problems.
Solution method: The libFastMesh (LFM) framework is based on a low-dissipation finite–volume discretization, together with high-order explicit time integration, for direct computation of aeroacoustic fields on unstructured meshes. Convective terms can be approximated using either the Kurganov–Noelle–Petrova (KNP) scheme or Pirozzoli's energy conserving approach. Time integration is fully–explicit and relies on a low-storage, 4th order Runge–Kutta scheme. The solver is specifically developed to run efficiently on modern many-core CPUs, and at large-scale where it exhibits excellent strong scaling features. This is made possible thanks to several optimizations that alleviate memory bandwidth and inter-node communication bottlenecks.
Additional comments: OpenFOAM-v2112+ is an installation prerequisite. Standard OpenFOAM pre-processing and post-processing tools, as well as mesh decomposition and reordering methods may be used with the solver.

View all citing articles on Scopus

View full text

Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations

Abstract

Appl. Numer. Math.

Appl. Numer. Math.

Appl. Math. Lett.

Comput. Math. Appl.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Appl. Numer. Math.

Appl. Numer. Math.

Comput. Math. Appl.

J. Comput. Phys.

Appl. Numer. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Appl. Numer. Math.

J. Comput. Phys.

Appl. Numer. Math.

Appl. Numer. Math.

On the removal of boundary errors caused by Runge–Kutta integration of nonlinear partial differential equations

SIAM J. Sci. Comput.

Does error control suppress spuriosity?

SIAM J. Numer. Anal.

Compressibility effects on the growth and structure of homogeneous turbulent shear flow

J. Fluid Mech.

Coefficients for the study of Runge–Kutta integration processes

J. Austral. Math. Soc. Ser. B

On Runge–Kutta processes of high order

J. Austral. Math. Soc. Ser. B

The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods

Fourth-order 2N-storage Runge–Kutta schemes, NASA TM-109112

Third-order 2N-storage Runge–Kutta schemes with error control, NASA TM-109111

The theoretical accuracy of Runge–Kutta time discretizations for the initial boundary value problem: a study of the boundary error

SIAM J. Sci. Comput.

A Kutta third-order procedure for solving differential equations requiring minimum storage

SIAM J. Numer. Anal.

A generalization of algebraic stability for Runge–Kutta methods

IMA J. Numer. Anal.

Generalized disks of contractivity for explicit and implicit Runge–Kutta methods, Report TRITA-NA-7906

Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations

Numerical Methods for Unconstrained Optimization and Nonlinear Equations

Numerical Methods for Differential Equations

Interpolants for Runge–Kutta formulas

ACM Trans. Math. Software

Two FORTRAN packages for assessing initial value methods

ACM Trans. Math. Software

A survey of the explicit Runge–Kutta method, Technical Report, 291/94

Interaction of a shock with a longitudinal vortex

J. Fluid Mech.

Low-order classical Runge–Kutta formulas with stepsize control and their application to some heat transfer problems, NASA Technical Report TR R-315

Economical evaluation of Runge–Kutta formulae

Math. Comp.

Computing optimal locally constrained steps

SIAM J. Sci. Statist. Comput.

ALGORITHM 611. Subroutines for unconstrained minimization using a model/trust region approach

ACM Trans. Math. Software

A process for the step-by-step integration of differential equations in an automatic digital computing machine

Proc. Cambridge Phil. Soc.

Total variation diminishing Runge–Kutta schemes

Math. Comp.