On gradient calculation in flux correction method
Matematičeskoe modelirovanie, Tome 31 (2019) no. 5, pp. 121-144.

Voir la notice de l'article provenant de la source Math-Net.Ru

Flux Correction method is a family of edge-based schemes for solving hyperbolic systems on unstructured meshes. The cruical operation there is a nodal gradient calculation of physical variables with at least second order of accuracy. There are two well-known procedures meeting this condition. One is based on Least Squares method and the other one is based on spectral elements. In this paper we compare resulting schemes and discuss their problems.
Keywords: unstructured mesh
Mots-clés : Flux Correction method, UFC scheme.
@article{MM_2019_31_5_a7,
     author = {P. A. Bakhvalov},
     title = {On gradient calculation in flux correction method},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {121--144},
     publisher = {mathdoc},
     volume = {31},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2019_31_5_a7/}
}
TY  - JOUR
AU  - P. A. Bakhvalov
TI  - On gradient calculation in flux correction method
JO  - Matematičeskoe modelirovanie
PY  - 2019
SP  - 121
EP  - 144
VL  - 31
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2019_31_5_a7/
LA  - ru
ID  - MM_2019_31_5_a7
ER  - 
%0 Journal Article
%A P. A. Bakhvalov
%T On gradient calculation in flux correction method
%J Matematičeskoe modelirovanie
%D 2019
%P 121-144
%V 31
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2019_31_5_a7/
%G ru
%F MM_2019_31_5_a7
P. A. Bakhvalov. On gradient calculation in flux correction method. Matematičeskoe modelirovanie, Tome 31 (2019) no. 5, pp. 121-144. http://geodesic.mathdoc.fr/item/MM_2019_31_5_a7/

[1] B. Stoufflet, J. Periaux, F. Fezoui, A. Dervieux, Numerical simulation of 3-D hypersonic Euler flows around space vehicles using adapted finite elements, AIAA Paper No 87-0560

[2] T.J. Barth, Numerical aspects of computing high reynolds number flows on unstructured meshes, AIAA Paper No 91-0721

[3] B. Koobus, F. Alauzet, A. Dervieux, “Numerical algorithms for unstructured meshes”, Computational Fluid Dynamics, ed. F. Magoules, CRC Press, 2011, 131–203 | DOI | Zbl

[4] I. Abalakin, P. Bakhvalov, T. Kozubskaya, “Edge-based reconstruction schemes for unstructured tetrahedral meshes”, International journal for numerical methods in fluids, 81:6 (2016), 331–356 | DOI | MR

[5] P. Bakhvalov, T. Kozubskaya, “EBR-WENO scheme for solving gas dynamics problems with discontinuities on unstructured meshes”, Comput. Fluids, 157 (2017), 312–324 | DOI | MR | Zbl

[6] I. Abalakin, V. Bobkov, V. Kozubskaya, “Implementation of the Low Mach Number Method for Calculating Flows in the NOISEtte Software Package”, Math. Mod. and Comp. Simul., 9:6 (2017), 689–697 | MR | Zbl

[7] A. Katz, V. Sankaran, “An efficient correction method to obtain a formally third-order accurate flow solver for node-centered unstructured grids”, J. Sci. Comput., 51:2 (2012), 375–393 | DOI | MR | Zbl

[8] B. Pincock, A. Katz, “High-order flux correction for viscous flows on arbitrary unstructured grids”, J. Sci. Comput., 61:2 (2014), 454–476 | DOI | MR | Zbl

[9] C.D. Work, A.J. Katz, Aspects of the flux correction method for solving the Navier-Stokes equations on unstructured meshes, AIAA paper No 2015-0834

[10] A. Katz, D. Work, “High-order flux correction/finite difference schemes for strand grids”, Journal of Computational Physics, 282 (2015), 360–380 | DOI | MR | Zbl

[11] O. Tong, Y. Yanagita, R. Shaap, S. Harris, A. Katz, High-Order strand grids methods for shock-turbulence interaction, AIAA paper No 2015-2283

[12] H. Nishikawa, “Accuracy-preserving Source Term Quadrature for Third-Order Edge-Based Discretization”, Journal of computational physics, 344 (2017), 595–622 | DOI | MR | Zbl

[13] P. Bakhvalov, T. Kozubskaya, “Modification of Flux Correction method for accuracy improvement on unsteady problems”, J. Comput. Phys., 338 (2017), 199–216 | DOI | MR

[14] P.A. Bakhvalov, “Realizatsya metoda korrektsii potokov na gibridnyh nestrukturirovannih setkah”, Keldysh Institute preprints, 2017, 038, 28 pp.

[15] T.J. Barth, A 3-D upwind Euler solver for unstructured meshes, AIAA Paper No 91-1548

[16] P. Bakhvalov, T. Kozubskaya, “Construction of Edge-Based 1-Exact Schemes for Solving the Euler Equations on Hybrid Unstructured Meshes”, Comp. Math. Math. Phys., 57:4 (2017), 680–697 | DOI | MR | Zbl

[17] H. Nishikawa, Beyond Interface Gradient. A General Principle for Constructing Diffusion Schemes, AIAA Paper No 2010-5093

[18] A.G. Kulikovsky, N. V. Pogorelov, A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Taylor Francis Inc., United States, 2000 | MR

[19] P.L. Roe, “Approximate Riemann solvers, parameter vectors, and difference schemes”, Journal of Computational Physics, 43:2 (1981), 357–372 | DOI | MR | Zbl

[20] H. Luo, J.D. Baumt, R. Lohner, “Edge-Based Finite Element Scheme for the Euler Equations”, AIAA Journal, 32:6 (1994) | DOI | Zbl

[21] P. Eliasson, EDGE, a Navier-Stokes solver for unstructured grids, Tech. Rep. FOI-R-0298-SE, FOI Swedish defence research agency, Division of Aeronautics, FFA, SE-172 90 STOCKHOLM, December 2001

[22] Y. Nakashima, N. Watanabe, H. Nishikawa, Hyperbolic Navier–Stokes solver for threedimensional flows, AIAA Paper No 2016-1101

[23] H. Nishikawa, Alternative formulations for first-, second-, and third-order hyperbolic Navier-Stokes schemes, AIAA Paper No 2015-2451

[24] C. Debiez, A. Dervieux, “Mixed-element-volume MUSCL methods with weak viscosity for steady and unsteady flow calculations”, Computers and Fluids, 29:1 (2000), 89–118 | DOI | MR | Zbl

[25] C. Debiez, A. Dervieux, K. Mer, B. Nkonga, “Computation of unsteady flows with mixed finite volume/finite element upwind methods”, International journal for numerical method in fluids, 27 (1998), 193–206 | 3.0.CO;2-D class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[26] B. Diskin, J.-L. Thomas, “Notes on accuracy of finite-volume discretization schemes on irregular grids”, Applied Numerical Mathematics, 60:3 (2010), 224–226 | DOI | MR | Zbl

[27] P.A. Bakhvalov, “O poryadke tochnosti reberno-orientirovannyh shem na setkah spetsial'nogo vida”, Keldysh Institute preprints, 2017, 079, 32 pp. | Zbl

[28] H. Nishikawa, “Divergence formulation of source term”, Journal of Computational Physics, 231 (2012), 6393–6400 | DOI | MR | Zbl

[29] L.G. Loitsyansky, Mechanics of Liquids and Gases, Pergamon Press, Oxford, UK, 1966, 804 pp. | MR

[30] E. Turkel, “Preconditioning Techniques in Computational Fluid Dynamics”, Annual Review of Fluid Mechanics, 31 (1999), 385–416 | DOI | MR

[31] H. Guillard, C. Viozat, “On the behaviour of upwind schemes in the low Mach number limit”, Computers and Fluids, 28:1 (1999), 63–86 | DOI | MR | Zbl

[32] S.V. Alekseenko, P.A. Kuibin, V.L. Okulov, Theory of Concentrated Vortices, Springer-Verlag, Berlin–Heidelberg, 2007, 487 pp. | MR | Zbl

[33] C. Geuzaine, J.-F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, 1997 | MR

[34] P.A. Bakhvalov, “Metod nestatsionarnogo korrektora dlya analiza tochnosti lineinyh poludiskretnyh shem”, Keldysh Institute preprints, 2018, 123, 38 pp.

[35] P. Woodward, P. Colella, “The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks”, Journal of computational physics, 54 (1984), 115–173 | DOI | MR | Zbl

[36] I.Y. Tagirova, A.V. Rodionov, “Application of the artificial viscosity for suppressing the carbuncle phenomenon in Godunov-type schemes”, Math. Models. Comput. Simul., 27 (2015), 47–64

[37] I.V. Abalakin, P.A. Bakhvalov, A.V. Gorobets, A.P. Duben, T.K. Kozubskaia, “Parallelnyi programmnyi kompleks NOISETTE dlia krupnomasshtabnykh raschetov zadach aerodinamiki i aeroakustiki”, Vychislitelnye metody i programmirovanie, 13 (2012), 110–125