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### Acceleration of nonlinear solvers for natural convection problems [article]

Sara Pollock, Leo G. Rebholz, Mengying Xiao
<span title="2020-04-14">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Set G(u, θ) = (G 1 (u, θ), G 2 (u, θ)) and G(w, z) = (G 1 (w, z), G 2 (w, z)).  ...  v) + (f, v), (3.26) b * (u, G 2 (u, θ), χ) + κ(∇G 2 (u, θ), ∇χ) = (γ, χ). = −b(h, G 1 (u + h, θ + s) − G 1 (u, s),g 1 ) + Ri( 0,g 2 T ,g 1 ) ≤ M ∇h ∇(G 1 (u + h, θ + s) − G 1 (u, θ) ∇g 1 + RiC 2 P ∇g  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.06471v1">arXiv:2004.06471v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nd6eptle7fatxhumrkcdh4dkte">fatcat:nd6eptle7fatxhumrkcdh4dkte</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200416025134/https://arxiv.org/pdf/2004.06471v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.06471v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Continuous Data Assimilation Reduced Order Models of Fluid Flow [article]

Camille Zerfas, Leo G. Rebholz, Michael Schneier, Traian Iliescu
<span title="2019-03-10">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend to lose accuracy on more complicated problems and over longer time intervals. Meanwhile, continuous data assimilation (DA) has recently been used to improve accuracy and, in particular, long time accuracy in fluid simulations by incorporating measurement
more &raquo; ... into the simulation. This paper synthesizes these two ideas, in an attempt to address inaccuracies in ROM by applying DA, especially over long time intervals and when only inaccurate snapshots are available. We prove that with a properly chosen nudging parameter, the proposed DA-ROM algorithm converges exponentially fast in time to the true solution, up to discretization and ROM truncation errors. Finally, we propose a strategy for nudging adaptively in time, by adjusting dissipation arising from the nudging term to better match true solution energy. Numerical tests confirm all results, and show that the DA-ROM strategy with adaptive nudging can be highly effective at providing long time accuracy in ROMs.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1903.04029v1">arXiv:1903.04029v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k3woppfgenhlxph2le6sbgtsoy">fatcat:k3woppfgenhlxph2le6sbgtsoy</a> </span>
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### Anderson-accelerated convergence of Picard iterations for incompressible Navier-Stokes equations [article]

Sara Pollock, Leo G. Rebholz, Mengying Xiao
<span title="2018-10-19">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
) − G(w)) 2 ≤ |b * (h, G(w), G(w + h) − G(w))| ≤ M ∇h ∇G(w) ∇(G(w + h) − G(w)) ≤ ν −1 M f −1 ∇h ∇(G(w + h) − G(w)) , thanks to (3.1) and (3.10) .  ...  Subtracting yields b * (w + h, G(w + h) − G(w), v) + b * (h, G(w), v) + ν(∇(G(w + h) − G(w)), ∇v) = 0. (3.11) Now setting v = G(w + h) − G(w) vanishes the first nonlinear term, and produces ν ∇(G(w + h  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1810.08494v1">arXiv:1810.08494v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i6jrf6wykjf5bjebvugbqqbcr4">fatcat:i6jrf6wykjf5bjebvugbqqbcr4</a> </span>
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### Conservation laws of turbulence models

Leo G. Rebholz
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7a65khxirfbfljbnqng5zzeslu" style="color: black;">Journal of Mathematical Analysis and Applications</a> </i> &nbsp;
(G N v · ∇G N v, AG N v) = (G N v · ∇G N v, G N v) = 0.  ...  (G N v · ∇G N v, ∇ × AG N v) = (G N v · ∇G N v, ∇ × G N v) = 1 2 ∇ (G N v) 2 , ∇ × G N v − G N v × (∇ × G N v), ∇ × G N v = 1 2 ∇ × ∇ (G N v) 2 , G N v − 0 = 0. The pressure term also vanishes.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jmaa.2006.02.026">doi:10.1016/j.jmaa.2006.02.026</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4kmiwoqypvdrbdjf3dg4cid4ty">fatcat:4kmiwoqypvdrbdjf3dg4cid4ty</a> </span>
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### Simple and efficient continuous data assimilation of evolution equations via algebraic nudging [article]

Leo G. Rebholz, Camille Zerfas
<span title="2018-10-08">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Briefly, define the matrix G = 1/2 −1 −1 5/2 , and note that G induces the norm x 2 G := (x, Gx), which is equivalent to the (L 2 ) 2 norm: C l x G ≤ x ≤ C u x G where C l = 3 − 2 √ 2 and C u = 3 + 2 √  ...  In all of our analysis, we invoke the G-norm and G-stability theory often used with BDF2 analysis, see e.g. [23, 10] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1810.03512v1">arXiv:1810.03512v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t47rz3gxxbdqtlmzf4phlbsszq">fatcat:t47rz3gxxbdqtlmzf4phlbsszq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191027164231/https://arxiv.org/pdf/1810.03512v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/42/d7/42d7dc5c8a9f771493ef25a68f9200cba4674680.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1810.03512v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### A multiscale V–P discretization for flow problems

Leo G. Rebholz
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xkk23xlywfez5l246cnmlit4qq" style="color: black;">Applied Mathematics and Computation</a> </i> &nbsp;
This paper gives a comprehensive numerical analysis of a multiscale method for equilibrium Navier Stokes equations. The method includes pressure regularization and eddy viscosity stabilizations both acting only on the finest scales. This method allows for equal order velocity-pressure spaces as well as the linear constant pair and the usual (P k , P k−1 ) pair. We show the method is optimal in a natural energy norm for all of these pairs of spaces, and provide guidance in choosing the
more &raquo; ... tion parameters. keywords : Navier Stokes, multiscale, subgrid eddy viscosity, pressure regularization, equal order interpolations AMS subject classifications: 65N12 76D05 65N15 2
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.amc.2005.10.030">doi:10.1016/j.amc.2005.10.030</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7xbwmnsuljbijonvcdly5wcdsu">fatcat:7xbwmnsuljbijonvcdly5wcdsu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20060830220410/http://www.math.pitt.edu/techrep/0509.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a0/77/a077ee1fdff325841ddd9491fb735cf88f459a12.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.amc.2005.10.030"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### Temporally regularized direct numerical simulation

William J. Layton, C. David Pruett, Leo G. Rebholz
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xkk23xlywfez5l246cnmlit4qq" style="color: black;">Applied Mathematics and Computation</a> </i> &nbsp;
Experience with fluid-flow simulation suggests that, in some instances, under-resolved direct numerical simulation (DNS), without a residual-stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales, may be as reliable as simulations based on more complex models of turbulence. One efficient and versatile manner to selectively damp under-resolved spatial scales is by a relaxation regularization, e.g. Stolz and
more &raquo; ... s [Phys. Fluids (1999)]. We consider the analogous approach based on time scales, time filtering and damping of under-resolved temporal features. The paper explores theoretical and practical aspects of temporally damped fluid flow simulations. We prove existence of unique strong solutions to the resulting continuum model. We also establish the effect of the damping of under-resolved temporal features as the energy balance and dissipation and prove that the time fluctuations → 0 in a precise sense to the temporally damped Navier-Stokes equations. The method is then demonstrated to obtain both steady-state and time-dependent coarse-grid solutions of the Navier-Stokes equations.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.amc.2010.05.031">doi:10.1016/j.amc.2010.05.031</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4mudksurmfc2xlbafrv4t6vcd4">fatcat:4mudksurmfc2xlbafrv4t6vcd4</a> </span>
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### An efficient algorithm for simulating ensembles of parameterized MHD flow problems [article]

Muhammad Mohebujjaman, Hongwei Wang, Leo G. Rebholz, Md. Abdullah Al Mahbub
<span title="2021-08-11">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Defining v j := u j + √ sB j , w j := u j − √ sB j , f 1,j := f j + √ s∇ × g j , f 2,j := f j − √ s∇ × g j , q j := p j + √ sλ j and r j := p j − √ sλ j produces the Elsässer variable formulation of the  ...  We assume the flow begins from rest, initially there is no magnetic field, and no external source is present in the system (i.e. f j = g j = 0).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.05110v1">arXiv:2108.05110v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3xmf35377fhbdkbjakbzot2cnu">fatcat:3xmf35377fhbdkbjakbzot2cnu</a> </span>
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### An efficient nonlinear solver and convergence analysis for a viscoplastic flow model [article]

Sara Pollock and Leo G. Rebholz and Duygu Vargun
<span title="2021-08-19">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Acknowledgments Authors Leo Rebholz and Duygu Vargun were partially supported by National Science Foundation grant DMS 2011490.  ...  Setg = G(u + h) − G(u) − G (u; h) for notational ease.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.08945v1">arXiv:2108.08945v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mz32gs3prrb6hi44pf7sesfy5y">fatcat:mz32gs3prrb6hi44pf7sesfy5y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210911003611/https://arxiv.org/pdf/2108.08945v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/19/47/1947f365ae32c1db6d28e257c70540536c78e9c6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.08945v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Efficient discretizations for the EMAC formulation of the incompressible Navier-Stokes equations [article]

Sergey Charnyi, Timo Heister, Maxim A. Olshanskii, Leo G. Rebholz
<span title="2017-12-03">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study discretizations of the incompressible Navier-Stokes equations, written in the newly developed energy-momentum-angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves
more &raquo; ... only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier-Stokes equations in primitive variables.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1712.00857v1">arXiv:1712.00857v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bn73zkoeund3bj2xee6xuarx2q">fatcat:bn73zkoeund3bj2xee6xuarx2q</a> </span>
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### Data-Driven Variational Multiscale Reduced Order Models [article]

Changhong Mou, Birgul Koc, Omer San, Leo G. Rebholz, Traian Iliescu
<span title="2020-08-10">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
i ≈ g(u r ) , ϕ i .(13) To determine the coefficients/parameters in g used in (13) , in the offline stage, we solve the following low-dimensional least squares problem: min g parameters M j=1 f (u F  ...  Once g is determined, the model (12) with the VMS-ROM closure term replaced by g yields the two-scale data-driven VMS-ROM (2S-DD-VMS-ROM): • u r , ϕ i = f (u r ) , ϕ i + g(u r ) , ϕ i , i = 1, . . . ,  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.06457v2">arXiv:2002.06457v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j6oone6cjvdaxly5bjbhjlpl54">fatcat:j6oone6cjvdaxly5bjbhjlpl54</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200823174545/https://arxiv.org/pdf/2002.06457v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/cb/49/cb497455e4032b03229d7871f30b175568481951.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.06457v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Modular grad-div stabilization for the incompressible non-isothermal fluid flows [article]

Mine Akbas, Leo G. Rebholz
<span title="2020-08-31">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
G. Rebholz. On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math., 40(2):491-516, 2014. [ 16 ] 16 V. John and A. Kindl.  ...  Rebholz. On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev., 59(3):492-544, 2017.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.10100v2">arXiv:2001.10100v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5rzqjqlh55bllcims6bvd3ogay">fatcat:5rzqjqlh55bllcims6bvd3ogay</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200903015506/https://arxiv.org/pdf/2001.10100v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.10100v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### The Reduced NS-α Model for Incompressible Flow: A Review of Recent Progress

Abigail L. Bowers, Leo G. Rebholz
<span title="2017-07-06">2017</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5bqnhgw7njc53i7otzqblwexk4" style="color: black;">Fluids</a> </i> &nbsp;
Bowers and Leo G. Rebholz did an equal amount of work on this manuscript, including the analysis, experiments, and writeup. Conflicts of Interest: The authors declare no conflict of interest.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/fluids2030038">doi:10.3390/fluids2030038</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z4lvwc22djhuzpvbxxk3e3v6t4">fatcat:z4lvwc22djhuzpvbxxk3e3v6t4</a> </span>
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### Enhanced Physics-Based Numerical Schemes for Two Classes of Turbulence Models

Leo G. Rebholz
<span title="">2009</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lrycfknfsbhqrmivcyowy76goa" style="color: black;">Advances in Numerical Analysis</a> </i> &nbsp;
These schemes are delicate extensions of a method created for the Navier-Stokes equations in Rebholz (2007), that achieve high physical fidelity by admitting balances of both energy and helicity that match  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2009/370289">doi:10.1155/2009/370289</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zvwytkcjkrfizcotw2z5pru7fu">fatcat:zvwytkcjkrfizcotw2z5pru7fu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190224205654/http://pdfs.semanticscholar.org/5dd2/c9f83680aee18f77e5620f0835c5c5f7548b.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5d/d2/5dd2c9f83680aee18f77e5620f0835c5c5f7548b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2009/370289"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> hindawi.com </button> </a>

### Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations [article]

Timo Heister and Maxim A. Olshanskii and Leo G. Rebholz
<span title="2015-11-25">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For the 2 × 2 matrix G := 1/2 −1 −1 5/2 , we introduce the G-norm χ G = (χ, Gχ), χ is vector valued. The G-norm is widely used in BDF2 analysis, see e.g. [4, 11] .  ...  The term χ 0 2 G can be decomposed as χ 0 2 G = (χ 0 , Gχ 0 ) = v 0 v 1 , 1/2 −1 −1 5/2 v 0 v 1 , and with some arithmetic we get that χ 0 2 G = Ω v 0 1 2 v 0 − v 1 + v 1 −v 0 + 5 2 v 1 dx = 1 2 v 1 2  ...  with g = 1, 0 T at the inflow, g = 0, 0 T on the walls, and with P 2 (τ h ) denoting the space of globally continuous functions which are quadratic on each triangle.  ...
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