NUMERICAL ANALYSIS OF AN ASYMPTOTIC MODEL FOR KAZHIKHOV-SMAGULOV TYPE EQUATIONS
Abstract
In this paper, we develop a hybrid scheme that combines finite volume and finite element methods for an asymptotic model of the Kazhikhov-Smagulov equations. We first establish the stability of the proposed scheme and its convergence toward the weak solution of the problem. Numerical simulations are then performed to verify the scheme's robustness and confirm the convergence of the solution.
2020 Mathematics Subject Classifications. Primary 35K51. Secondary 35Q30.
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S.N. Antontsev, A.V. Kazhikhov and V.N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, Studies in Mathematics and Its Applications, 22, North-Holland, Publishing Co., Amesterdam, 1990.
S. Boivin and J.M. H ́erard Un sch ́ema de type volumes finis pour r ́esoudre les ́equations de Navier-Stokes sur une triangulation.Revue europ ́eenne des ́el ́ements finis. Volume 5- n° 4/1996, pages 461 `a 490
R.C. Cabrales, F. Guill`en-Gonz`alez and J.V. Guti ́errez-Santacreu, Stability and convergence for a complete model of mass diffusion, Applied Numerical Mathematics 61 (2011), 1161-1185.
C. Calgaro, C. Colin, E. Creus ́e. A combined Finite Volumes -Finite Elements method for a low-Mach model. International Journal for Numerical Methods in Fluids, 2019, 90 (1), pp.1-21.
1002/fld.4706. hal-01574894v2
C. Calgaro, E. Creus ́e, and T. Goudon. An hybrid finite volume-finite element method for
variable density incompressible flows.J. Comput. Phys., 227(9):4671-4696, 2008.
C. Calgaro and M. Ezzoug. L∞-stability of IMEX-BDF2 finite volume scheme for convection- diffusion equation. FVCA 2017: Finite Volumes for Complex Applications VIII - Methods and
Theoretical Aspects, Jun 2017, Lille, France. pp.245-253
C. Calgaro; M. Ezzoug and E. Zahrouni. Stability and convergence of an hybrid finite volume-
finite element method for a multiphasic incompressible fluid model, Commun. Pure Appl.
Anal., 17(2) :429-448, 2018.
P. G. Ciarlet. The finite element method for elliptic problems. North-Holland Publishing Co.,
Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4.
R. Eymard, T. Gallouet and R. Herbin, Finite volume methods,Handbook of Numerical Anal-
ysis, vol. VII, North-Holland, Amsterdam, (2000), 7131020.
M. Feistauer, J. Felcman and M. Luk`a ̆cov`a-Medvid’ov`a, On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems,.Numerical Methods Partial Differential Equations 13 (1997), 163(190).
F. Guill`en-Gonz`alez and J.V. Guti ́errez-Santacreu. Error estimates of a linear decoupled Euler-FEM scheme for a mass diffusion model., Numer. Math. 117 (2011), 333-371.
F. Guili ́en-Gonz`alez, P. Dam`azio and M.A. Rojas-Medar, Approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion, ,J. Math. Anal. Appl. 326 (2007), 468-487.
F. Guill ́en-Gonza`lez and J.V. Guti`errez-Santacreu, Unconditional stability and convergence of fully discrete schemes for 2D viscous fluids models with mass diffusion, Mathematics of Computation. 77 (263) (2008), 1495-1524.
F. Guill`en-Gonz`alez and J.V. Guti`errez-Santacreu, Conditional stability and convergence of fully discrete scheme for three-dimensional Navier-Stokes equations with mass diffusion, SIAM J. Numer. Anal. 46 (5) (2008), 2276-2308.
J.L. Lions, Quelques m ́ethodes de r ́esolution des proble`mes aux limites non lin ́eaires, Dunod, Gauthier-Villars, Paris, 1969.
T. Ngom, A. Ouedraogo and M. Dahi Asymptotique Devlopment of Kazhikhov-Smagulov Equations. Bulletin of Mathematical Analysis and Applications. Volume 16 Issue 4(2024), Pages 35-60 https://doi.org/10.54671/BMAA-2024-4-3
D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cam- bridge University Press, 2003.
J. Simon Compacts sets in Lp(0,T;B)Ann. Mat. Pura Appl. 146 (1987), 65(96).
R. Teman, Navier-Stokes equations, theory and numerical analysis.Revised Edition, Studies in mathematics and its applications vol. 2, North Holland Publishing Company-Amsterdam, New York, 1984.
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