Assessment of selfadapting local projectionbased solvers for laminar and turbulent industrial flows
 Tomás Chacón Rebollo^{1, 2},
 Enrique Delgado Ávila^{1, 2}Email authorView ORCID ID profile,
 Macarena Gómez Mármol^{1} and
 Samuele Rubino^{1, 2}
https://doi.org/10.1186/s1336201800454
© The Author(s) 2018
Received: 15 March 2018
Accepted: 29 May 2018
Published: 8 June 2018
Abstract
In this work, we study the performance of some local projectionbased solvers in the Large Eddy Simulation (LES) of laminar and turbulent flows governed by the incompressible Navier–Stokes Equations (NSE). On one side, we focus on a highorder termbyterm stabilization Finite Element (FE) method that has one level, in the sense that it is defined on a single mesh, and in which the projectionstabilized structure of standard Local Projection Stabilization (LPS) methods is replaced by an interpolationstabilized structure. The interest of LPS methods is that they ensure a selfadapting high accuracy in laminar regions of turbulent flows, which turns to be of overall optimal high accuracy if the flow is fully laminar. On the other side, we propose a new Reduced Basis (RB) Variational MultiScale (VMS)Smargorinsky turbulence model, based upon an empirical interpolation of the subgrid eddy viscosity term. This method yields dramatical improvements of the computing time for benchmark flows. An overview about known results from the numerical analysis of the proposed methods is given, by highlighting the used mathematical tools. In the numerical study, we have considered two well known problems with applications in industry: the (3D) turbulent flow in a channel and the (2D/3D) recirculating flow in a liddriven cavity.
Keywords
1 Introduction
In this paper, we consider two different FE approximations of the NSE arising from local projectionbased methods for the LES of laminar and turbulent incompressible flows. The interest of the presented projectionbased methods is that they allow an important reduction on the computational time requirements with respect to classical methodologies, providing at the same time highorder accuracy with reduced computational complexity. First, we introduce a full order model, which is a variant of standard LPS schemes, for the evolution NSE. The most relevant feature from the practical point of view is that the proposed full order approach looks simple overall, yet it manages to solve complex high Reynolds numbers flows on relatively coarse grids. Then, to further reduce computational complexity, we also consider a reduced order model, which consists of a RB VMSSmagorinsky turbulence model, for the steady NSE (its extension to evolution NSE through its combination with a proper orthogonal decomposition strategy is today in progress). The most relevant feature from the practical point of view is that the proposed reduced order approach yields a dramatic speedup of the computing time with respect to the corresponding high fidelity model, while maintaining a similar accuracy up to moderate Reynolds numbers.
On the one hand, we focus on the highorder termbyterm stabilization method introduced in (cf. [1]) for the Oseen equations. This method is developed by a purely numerical approach that does not require any adhoc eddy viscosity. It is a particular type of LPS scheme, which constitutes a lowcost, accurate solver for incompressible flows, despite being only weakly consistent since it does not involve the full residual. It differs from the standard LPS methods (cf. [2, 3]) because it uses continuous buffer functions, it does not need enriched FE spaces, it does not need elementwise projections satisfying suitable orthogonality properties, and it does not need multiple meshes. Commonly to standard LPS methods, the stabilization terms only act on the small scales of the flow, thus ensuring a higher accuracy with respect to more classical stabilization procedures, such as penaltystabilized methods (cf. [4]). This method has been recently supported by a thorough numerical analysis (existence and uniqueness, stability, convergence, error estimates, asymptotic energy balance) for the nonlinear problem related to the evolution NSE (cf. [5, 6]), using a semiimplicit Euler scheme for the monolithic discretization in time. The main results from the numerical analysis of the proposed LPS method will be recalled here. We will also focus on an efficient time discretization of this method via a stable velocitypressure segregation, using semiimplicit Backward Differentiation Formulas up to the second order (BDF2), with a special emphasis on its numerical solution in a parallel setting (cf. [7]). We show some relevant 3D numerical tests, to assess the performance of the proposed LPS method as an efficient and accurate solver for the simulation of laminar and turbulent incompressible complex flows that could arise in industrial applications.
On the other hand, we present a promising RB VMSSmagorinsky turbulence model, based upon the approximation of the subgrid eddy viscosity term by means of the empirical interpolation method, and on the approximation of velocitypressure by a Greedy algorithm built with a specific error estimator. The numerical analysis for the steady NSE is performed. Also, we present some numerical results for the benchmark 2D liddriven cavity flow problem that show a dramatic speedup of the computing time. The adaptation of this solver to complex flows, now in progress, is of primary interest for analysis and optimal design in fluid mechanics industrial applications.
The paper is organized as follows. In Sect. 2, we describe the proposed LPS approximation of the incompressible evolution NSE, commonly referred as highorder termbyterm stabilization, and we state its main properties. After recalling the main results from the numerical analysis of the proposed LPS method, we present an efficient and accurate time discretization of this model by means of an incremental pressurecorrection algorithm with semiimplicit BDF2, and describe the parallel solver developed for the fully discrete problem. In Sect. 3, we show numerical studies to assess the performance of the proposed LPS strategy. In Sect. 4, the proposed RB VMSSmagorinsky model for the steady NSE is introduced and theoretically analyzed. Numerical studies for this model are carried out in Sect. 5. In particular, the high computational efficiency of the proposed RB VMSSmagorinsky model is showcased. Finally, Sect. 6 states the main conclusions of the paper.
2 A highorder LPS discretization of evolution NSE
We introduce a numerical approximation for an InitialBoundary Value Problem (IBVP) describing the incompressible evolution NSE. For the sake of simplicity, we just impose homogeneous Dirichlet boundary condition on the whole boundary. More general inflow boundary conditions may be taken into account by standard lifting techniques for NSE. Also, the treatment of general nonlinear wall law boundary conditions may be found in [8].
Let \([0,T]\) be the time interval, and Ω a bounded polyhedral domain in \(\mathbb{R}^{d}\), \(d=2\) or 3, with a Lipschitzcontinuous boundary \(\Gamma =\partial \Omega \). Let \(\{{\mathcal {T}}_{h}\}_{h>0}\) be a family of affineequivalent, conforming (i.e., without hanging nodes) and regular triangulations of Ω̅, formed by triangles (\(d=2\)) or tetrahedra (\(d=3\)).

Initialization. Set: \(\mathbf{u}_{h}^{0}=\mathbf{u}_{0h}\).

Iteration. For \(n=0,1,\ldots,N1\): Given \(\mathbf{u}_{h}^{n}\in {\mathbf {X}}_{h}\), find \((\mathbf{u}_{h}^{n+1},p_{h}^{n+1})\in {\mathbf {X}}_{h}\times {\mathbb {M}}_{h}\) such that:for any \((\mathbf{v}_{h},q_{h}) \in {\mathbf {X}}_{h}\times {\mathbb {M}}_{h}\), where \({\mathbb {M}}_{h}=Y_{h}^{l}\cap L_{0}^{2}(\Omega)\), \(\overline{\mathbf{f}}^{n+1}\) is the average value of f in \([t_{n},t_{n+1}]\), and \(\mathbf{u}_{0h}\) is some stable approximation to \(\mathbf{u}_{0}\) belonging to \({\mathbf {X}}_{h}\), e.g., its discrete Stokes projection.$$ \textstyle\begin{cases} (\frac{\mathbf{u}_{h}^{n+1}\mathbf{u}_{h}^{n}}{\Delta t},\mathbf{v}_{h})_{\Omega }+b(\mathbf{u}_{h}^{n},\mathbf{u}_{h}^{n+1},\mathbf{v}_{h})+a(\mathbf{u}_{h}^{n+1},\mathbf{v}_{h}) \\ \quad{}(p_{h}^{n+1},\nabla\cdot \mathbf{v}_{h})_{\Omega }+s_{\mathrm{conv}}(\mathbf{u}_{h}^{n},\mathbf{u}_{h}^{n+1},\mathbf{v}_{h})+s_{\mathrm{div}}(\mathbf{u}_{h}^{n+1},\mathbf{v}_{h}) = \langle \overline{\mathbf{f}}^{n+1}, \mathbf{v}_{h}\rangle, \\ (\nabla\cdot \mathbf{u}_{h}^{n+1},q_{h})_{\Omega }+s_{\mathrm{pres}}(p_{h}^{n+1},q_{h}) = 0, \end{cases} $$(2)
2.1 Numerical analysis
The discrete method (2) has been recently supported by a thorough numerical analysis (stability, convergence, error estimates, asymptotic energy balance) for the nonlinear problem related to the evolution NSE (cf. [5, 6]), which is to our knowledge unavailable for most turbulence models in the current literature (cf. [13]).
Here, we recall the main results obtained from the numerical analysis. First, we need the following technical hypothesis on the stabilization coefficients:
Hypothesis 2.1
We next state a specific discrete infsup condition for the stabilized approximation that is essential for the stability of method (2).
Lemma 2.1
The proof of this lemma can be derived from [1], where it is also shown that the discrete infsup condition (9) can be extended to a more complex condition that holds for a simply regular family of triangulations.

\(\mathbf{u}_{h}\) is the piecewise linear in time function with values on \({\mathbf {X}}_{h}\) such that \(\mathbf{u}_{h}(t_{n})=\mathbf{u}_{h}^{n}\),

\(\widetilde{p_{h}}\) is the piecewise constant in time function that takes the value \(p_{h}^{n+1}\) on \((t_{n},t_{n+1})\),

\(P_{h}(t)=\int_{0}^{t}\widetilde{p_{h}}(s) \,ds\).
Theorem 2.1
The convergence of method (2) is now stated as follows:
Theorem 2.2
Assume that Hypothesis 2.1 holds, and let \(\mathbf{f}\in L^{2}(\mathbf{H}^{1})\), \(\mathbf{u}_{0}\in\mathbf{L}^{2}\). Then, the sequence \(\{(\mathbf{u}_{h},P_{h})\}_{h>0}\) contains a subsquence \(\{( \mathbf{u}_{h'},P_{h'})\}_{h'>0}\) that is weakly convergent in \(L^{2}(\mathbf{H}^{1})\times L^{2}(L^{2})\) to a weak solution \((\mathbf{u},P)\) of the unsteady NSE, being P the time primitive of the physical pressure. Moreover, \(\{\mathbf{u}_{h'}\}_{h'>0}\) is weakly^{∗} convergent in \(L^{\infty}(\mathbf{L}^{2})\) to u, strongly in \(L^{2}(\mathbf{H}^{s})\) for \(0\leq s<1\), and \(\{P_{h'}\}_{h'>0}\) is weakly^{∗} converegent in \(L^{\infty}(L^{2})\) to P. If the weak solution of the unsteady NSE is unique, then the whole sequence converges to it.
The proofs of these theorems can be directly derived by the ones performed in [14].
We now state the following error estimate result:
Theorem 2.3
A detailed proof of this theorem can be found in [5]. Taking \(s=l\), if the flow is regular enough, we obtain convergence of optimal order, and the order decreases with the regularity.
Remark 2.1
The proof of Theorem 2.3, that implies more concretely a strong convergence result for solutions with slightly increased regularity (it is sufficient \((\mathbf{u},p)\in C^{0}(\mathbf{H}^{2})\times C^{0}(H^{1})\), even if the convergence order in space is limited to one, due to the pressure stabilizing term), contains as a subproduct the asymptotic energy balance of the approximation (2): the total energy balance is asymptotically maintained in such a way that the subgrid energy due to stabilizing terms asymptotically vanish (see [15], Sect. 3.4). This is not the case if we consider the natural minimal regularity of the continuous solution: indeed, due to the low regularity of the weak solution, we can just prove an energy inequality, due to the dissipative nature of the approximation (2), by using that the subgrid stabilizing energy terms are positive (cf. [14]).
Remark 2.2
The presented analysis for the proposed highorder termbyterm stabilization procedure has been extended to geophysical flows governed by the primitive equations of the ocean [16] and buoyant flows governed by the Boussinesq equations [17]. Also, it has been combined with a Variational MultiScale (VMS)Smagorinsky term and wall laws for the accurate simulation of turbulent boundary layers in [14, 15, 18].
2.2 An efficient time discretization of the NSE with LPS modeling in a HPC framework
In practical implementations, for the first time step (\(n=0\)) we use a BDF1 scheme (\(r=1\)) to initialize the algorithm with \(\widetilde{\mathbf{u}}_{h}^{0}=\mathbf{u}_{h}^{0}\) and \(p_{h}^{0}\) some stable approximations to \(\mathbf{u}^{0}\) and \(p^{0}\), respectively. Note that this scheme coincides with the semiimplicit Euler method (2). Then, a BDF2 scheme (\(r=2\)) is applied for \(n\geqslant1\).
The semiimplicit discretization in time segregating velocity and pressure through a standard incremental timesplitting helps to construct an efficient linear solver to the NSE system for the LES of laminar and turbulent incompressible flows. In the first step (17), a convectiondominated convection–diffusion–reaction subproblem for the intermediate velocity must be solved. The second step (18) consists of a stabilized pressurePoisson subproblem. Solving the associated large linear systems could become extremely expensive from the computational point of view, that is why we adopt in the numerical implementation a highly parallel strategy based on Domain Decomposition Methods (DDM). Both steps are solved by using a DDM preconditioner with the GMRES iterative method applied to the associated system in the parallel framework described in [7], and a convincing strong scaling analysis of the used algorithm is showcased in this framework. In particular, we have interfaced the proposed fully discrete scheme (17)–(18) with HPDDM [20, 21], a high performance unified framework for DDM, and used a parallel iterative linear solver based on an optimized Schwarz DDM as preconditioner [20, 22]. In this manner, we obtain an efficient, i.e., robust and fast, solver for the High Performance Computing (HPC) of laminar and turbulent incompressible flows in the opensource FE software FreeFem++ interfaced with the library HPDDM. The proposed parallel strategy is tested for the recirculating flow in a 3D liddriven cavity in the next section.
3 Numerical experiments with LPS by interpolation method
In this section, we discuss some numerical results to analyze the numerical performances of the proposed LPS model applied to the computation of laminar and turbulent complex flows that could arise in industrial applications, also on massive parallel settings.
3.1 Turbulent channel flow (3D)
We present results of a fully developed turbulent flow in a 3D channel at moderate friction Reynolds number \(\mathit{Re}_{\tau}=180\). The 3D channel flow is one of the most popular test problems for the investigation of wall bounded turbulent flows, whereas turbulent boundary layers are of high practical relevance in aerodynamics industries. The proposed test consists of a fluid that flows between two parallel walls driven by an imposed pressure gradient source term which is defined by the friction Reynolds number \(\mathit{Re}_{\tau}\). For the setup of our numerical simulations, we chose to follow the guidelines given by Gravemeier in [23]. As a benchmark, we will use the fine Direct Numerical Simulation (DNS) of Moser, Kim and Mansour [24].
The boundary conditions are periodic in both the streamwise and spanwise directions (homogeneous directions). We perform a comparison between the application of the logarithmic walllaw of Prandtl and Von Kármán and noslip boundary conditions at the walls.
We aim to obtain a good accuracy with a relatively coarse spatial resolution. The computational grid consists of a \(16\times32\times16\) partition of the channel, uniform in the homogeneous directions. The distribution of nodes in the wallnormal direction is nonuniform, and obeys the cosine function of Gauss–Lobatto. We use 3D \(\mathbb{P}_{2}\) FE for velocity and pressure.
We consider for this test a semiimplicit Crank–Nicolson scheme for the monolithic temporal discretization. This provides a good compromise between accuracy and computational complexity, while keeping the numerical diffusion levels below the subgrid terms (cf. [25]). Indeed, on the one side, it produces less numerical diffusion with respect to a simple semiimplicit Euler scheme, and thus it does not tend to artificially increment the turbulent diffusion. On the other side, despite being a firstorder method, it already provides accurate results at the considered moderate friction Reynolds number \(\mathit {Re}_{\tau }\), being less expensive in terms of storage requirements with respect to the twostep BDF2 scheme described in Sect. 2.2, which instead allows to achieve a secondorder accuracy in time.
Figure 1 (right) displays the normalized r.m.s. values of the streamwise velocity fluctuations. If we compare with DNS data the LPS method by interpolation tested with noslip boundary conditions, we observe a noticeable overprediction, which seems to be corrected by the use of wall laws in the socalled inertial layer, starting from the first interior node.
Remark 3.1
Note that the present numerical study differs from the one performed in [14, 15, 18], where the combination with a VMSSmagorinsky turbulence model has been considered on a computational grid that consists of a 16^{3} partition of the channel. The presented results show that taking into account just a purely LPS method (no adhoc eddy viscosity of Smagorinskytype is introduced) provides almost the same highorder accuracy of the more complex VMSLPS method in [14, 15, 18], whenever we consider a proper refinement just on the wallnormal direction, giving results very close to the fine DNS.
3.2 Liddriven cavity flow (3D)
In this section, the 3D liddriven cavity test is performed to investigate the numerical performances of the proposed solver at laminar, transient, and turbulent regimes, also on massive parallel settings. The liddriven cavity flow is one of the most studied problem in Computational Fluid Dynamics (CFD), that exhibits one direction of inhomogeneity. This problem is characterized by a fluid flow in a cubic domain driven by a tangential unitary velocity along one of the six boundary surfaces. Homogeneous Dirichlet conditions are adopted on all the other boundaries.
The recirculating flow in a 3D liddriven cavity presents the occurrence of some considerable 3D features, even at relatively low Reynolds numbers. One of the most remarkable is the formation of Taylor–Görtlerlike (TGL) vortices at the corners of the bottom of the cavity. Small counterrotating vortices are formed as a result of the curvature of the streamlines due to the main vortex in the middle of the cavity. Following the work of Gravemeier et al. [26], we simulate the 3D cavity flow at Reynolds numbers \(\mathit{Re}=3200, 7500, 10\text{,}000\), to cover respectively the laminar, transient and turbulent regimes.
Also for this test, we first aim to obtain a good accuracy with a relatively coarse spatial resolution. The computational grid consists of a 32^{3} partition of the unit cube, uniform in the ydirection, and refined towards the walls in both x and zdirections using the hyperbolic tangent function, in order to handle large velocity gradients. Again, we use 3D \(\mathbb{P}_{2}\) FE for velocity and pressure. For this test, we apply the efficient time discretization described in Sect. 2.2 in a parallel setting. Indeed, since we arrive to high Reynolds numbers, then the use of (at least) secondorder accurate discretization in time has been found to be essential in order to achieve a reasonable accuracy. The results are graphically compared to the experimental data of Prasad and Koseff [27], and numerical results of Gravemeier et al. [26], obtained by a threelevel VMSSmagorinsky method (VMS3L).
Example 3.2. Number of GMRES iterations of the solver for \(\mathit{Re}=10\text{,}000\)
# of subdomains  # of velocity iterations  # of pressure iterations 

1024  18  17 
2048  19  12 
4096  21  14 
8192  24  15 
16,384  23  13 
Remark 3.2
Note that this numerical study differs from the one performed in [7], where just mixed infsup stable FE of Taylor–Hood type (\(\mathbb{P}_{2}/\mathbb{P}_{1}\)) have been considered for the pair velocity/pressure in numerical experiments, for which the pressure stabilized term (7) is neglected. The presented results with equalorder \(\mathbb{P}_{2}/\mathbb{P}_{2}\) FE considering pressure stabilization are almost comparable with the ones of [7], thus being in good agreement with experimental data. However, the use of mixed FE leads to cheaper (amortized setup) Poisson solves for the pressure equation (18) in the HPC framework considered. This is reflected in Fig. 6, where we start to observe from 4096 processes that using equalorder FE leads to a slight deterioration in the scalability of the total average time to complete a time step, which is not the case when considering mixed FE in [7]. This follows from the fact that using equalorder FE requires the assembly of an additional term for pressure stabilization and, as consequence, this results in an increased computational cost with respect to mixed formulations. Nevertheless, the number of GMRES iterations remains stable, and in the same low range as in Ref. [7] (see Table 1). To sum up, also the parallel performances in the case of equalorder FE are rather satisfactory, and seem to be in accordance with the current stateoftheart, e.g., [28].
4 RB VMSSmagorinsky model
In this section, we present a RB VMSSmagorinsky turbulence model. This differs from the reduced order model considered in [29, 30], which is just based on the simpler Smagorinsky turbulence model. In particular, an interpolation operator has been introduced, in order to restrict the influence of the eddy viscosity just to the small resolved scales. This allows to avoid the overdiffusion phenomenon of the standard Smagorinsky model, where the effect of the unresolved scales is typically taken into account equally for all resolved flow scales, and as a consequence, the large scales are usually overdamped, yielding results with lower accuracy, unuseful for most flows of practical interest in industry.
The idea supporting the RB method is to build a reduced basis formed by a few number of solutions from the original problem for some values of the parameter, in the offline phase. Then, the problem is solved by a Galerkin projection onto the space \({\mathbf {X}}_{N}\times {\mathbb {M}}_{N}\) spanned by the RB, in the online phase.
The eddy viscosity in the VMSSmagorinsky model is defined as \(\nu _{T}(\mathbf{u}_{h}')= (C_{S} h_{K})^{2} \nabla \mathbf{u}'_{h_{K}} \), where \(\cdot \) denotes the Frobenius norm and \(C_{S}\) is the Smagorinsky constant. To linearise this nonlinear term, we use the Empirical Interpolation Method (cf. [32]). This allows a large speedup in the solution of the RB problem.
Remark 4.1
Problem (22) is supposed to have homogeneous Dirichlet boundary conditions. In the case of considering nonhomogeneous Dirichlet boundary conditions, problem (22) is transformed to an equivalent one with homogeneous boundary conditions by considering a lift function. For more details, see e.g. [30, 33].
4.1 A posteriori error bound estimator
Then, according to the BRR theory (cf. [34, 35]), it will follow that in a neighbourhood of \(U_{h}(\mu)\) the solution of (22) is unique and bounded in \(\Vert {\cdot} \Vert _{\overline{\mathbf{X}}_{h}}\) in terms of the data. The proof of the existence of \(\beta_{h}(\mu)\) can be derived from Proposition 4.2 of [30], thanks to the fact that the interpolation operator \(\Pi_{h}\) satisfies optimal error estimates (cf. [31]).
For the development of the a posteriori error bound, we start by proving that the directional derivative of the operator \(A(\cdot,\cdot;\mu)\) is globally lipschitz. The proof of the following Lemma can be derived from Lemma 5.1 in [30].
Lemma 4.1
The suitability of the a posteriori error bound estimator is stated by the following results. Their proofs can be derived from [30], taking into account that the interpolation operator \(\Pi_{h}\) is uniformly stable in \(H^{1}\)norm.
Theorem 4.1
Theorem 4.2
5 Numerical experiments with RB VMSSmagorinsky method
In order to implement the VMSSmagorinsky eddy diffusion, we consider a standard nodal Lagrange interpolation operator for its simplicity and efficiency with respect to other choices. First, to start the Greedy algorithm, we set up the Empirical Interpolation Method in order to approximate the eddy viscosity term. We need 34 basis functions until reaching the tolerance for the error in the Empirical Interpolation Method, see [30] for more details.
We start the Greedy algorithm for \(\mu=1000\), and we need \(N=9\) basis functions until reaching the Greedy tolerance of 10^{−5} for the a posteriori error bound estimator. Since the VMSSmagorinsky model is less diffusive than the classical Smagorinsky model, the number of basis functions needed for the RB model is lower. In [30] where a classical Smagorinsky model is considered, for the same numerical test, the Greedy algorithm selects 12 basis functions instead of 9 basis functions.
Computational time for FE solution and RB online phase, with the speedup and the relative error
Data  μ = 1620  μ = 2142  μ = 3693  μ = 4745 

\(T_{\text{FE}}\)  1486.4 s  1972.1 s  3089.3 s  3777.51 s 
\(T_{\text{online}}\)  0.51 s  0.52 s  0.52 s  0.52 s 
speedup  2869  3773  5935  7264 
\(\\mathbf{u}_{h}\mathbf{u}_{N}\_{{\mathbf {X}}_{h}}\)  1.9⋅10^{−6}  1.58⋅10^{−6}  2.62⋅10^{−6}  4.99⋅10^{−6} 
\(\p_{h}p_{N}\_{{\mathbb {M}}_{h}}\)  2.94⋅10^{−7}  1.79⋅10^{−7}  1.11⋅10^{−7}  1.25⋅10^{−7} 
6 Conclusions
The numerical studies performed in the present paper indicate that the considered LPS method is able to reproduce first and secondorder statistics up to a turbulent regime for relatively coarse meshes, with a similar (or even higher) accuracy than a more complex VMSLES method [26]. We studied the parallel performances of the proposed solver implemented in a HPC framework, showing rather good scalability results up to thousands of cores. This promotes the present method as a suitable and useful tool in the challenging simulation of turbulent flows, since providing reliable numerical results with a comparatively small computational complexity, which is an extremely important feature in the context of realistic industrial applications in CFD. We also presented a RB VMSSmagorinsky model, for which we developed an a posteriori error bound estimator, and we presented a numerical test in which we showed a speedup of several thousands in the computation of the numerical solution of the VMSSmagorinsky model. We thus enhanced the results presented in [30], considering a more accurate highorder method, with higher speedup in the computation of the RB solution in the online phase.
Declarations
Acknowledgements
The research of the authors has been partially supported by the Spanish Government Project MTM201564577C21R.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Authors’ information
Not applicable.
Funding
The research of the authors has been partially supported by the Spanish Government Project MTM201564577C21R.
Authors’ contributions
All authors contributes to this paper as a whole. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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