A Characteristics-Mix Stabilized Finite Element Method for Variable Density Incompressible Navier-Stokes Equations
Affiliation(s)
1 Kunming Medical University Haiyuan College, Kunming, China. 2 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China. 3 School of Computer Engineering and Science, Shanghai University, Shanghai, China.
1 Kunming Medical University Haiyuan College, Kunming, China. 2 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China. 3 School of Computer Engineering and Science, Shanghai University, Shanghai, China.
ABSTRACT
This paper describes a characteristics-mix finite element method for the computation of incompressible Navi-er-Stokes equations with variable density. We have introduced a mixed scheme which combines a characteristics finite element scheme for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint. The proposed method has a lot of attractive computational properties: parameter-free, very flexible, and averting the difficulties caused by the original equations. The stability of the method is proved. Finally, several numerical experiments are given to show that this method is efficient for variable density incompressible flows problem.
This paper describes a characteristics-mix finite element method for the computation of incompressible Navi-er-Stokes equations with variable density. We have introduced a mixed scheme which combines a characteristics finite element scheme for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint. The proposed method has a lot of attractive computational properties: parameter-free, very flexible, and averting the difficulties caused by the original equations. The stability of the method is proved. Finally, several numerical experiments are given to show that this method is efficient for variable density incompressible flows problem.
KEYWORDS
Characteristic Finite Element, Variable Density Flows, Naiver-Stokes Equations, Stabilized Method
Characteristic Finite Element, Variable Density Flows, Naiver-Stokes Equations, Stabilized Method
Cite this paper
Xiong, F. , Mei, L. , Li, Y. and Zhang, W. (2015) A Characteristics-Mix Stabilized Finite Element Method for Variable Density Incompressible Navier-Stokes Equations. Advances in Pure Mathematics, 5, 251-266. doi: 10.4236/apm.2015.55026.
Xiong, F. , Mei, L. , Li, Y. and Zhang, W. (2015) A Characteristics-Mix Stabilized Finite Element Method for Variable Density Incompressible Navier-Stokes Equations. Advances in Pure Mathematics, 5, 251-266. doi: 10.4236/apm.2015.55026.
References
[1] Guermond, J.L. and Salgado, A. (2009) A Splitting Method for Incompressible Flows with Variable Density Based on a Pressure Poisson Equation. Journal of Computational Physics, 228, 2834-2846.
http://dx.doi.org/10.1016/j.jcp.2008.12.036 [2] Pyo, J.H. and Shen, J. (2007) Gauge-Uzawa Methods for Incompressible Flows with Variable Density. Journal of Computational Physics, 221, 181-197.
http://dx.doi.org/10.1016/j.jcp.2006.06.013 [3] Guermond, J.L. and Quartapelle, L. (2000) A Projection FEM for Variable Density Incompressible Flows. Journal of Computational Physics, 165, 167-188.
http://dx.doi.org/10.1006/jcph.2000.6609 [4] Chorin, A.J. (1968) Numeriacl Solution of the Navier-Stokes Equations. Mathematics of Computation, 22, 745-762.
http://dx.doi.org/10.1090/S0025-5718-1968-0242392-2 [5] Chorin, A.J. (1969) On the Convergence of Discrete Approximations to the Navier-Stokes Equations. Mathematics of Computation, 23, 341-353. http://dx.doi.org/10.1090/S0025-5718-1969-0242393-5 [6] Temam, R. (1977) Navier-Stokes Equations, Studies in Mathematics and Its Applications 2. North-Holland, Amsterdam. [7] Temam, R. (1969) Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II). Archive for Rational Mechanics and Analysis, 33, 377-385.
http://dx.doi.org/10.1007/BF00247696 [8] Guermond, J.L., Minev, P. and Shen, J. (2006) An Overview of Projection Methods for Incompressible Flows. Computer Methods in Applied Mechanics and Engineering, 195, 6011-6045.
http://dx.doi.org/10.1016/j.cma.2005.10.010 [9] Fraigneau, Y., Guermond, J.L. and Quartapelle, L. (2001) Approximation of Variable Density Incompressible Flows by Means of Finite Elements and Finite Volumes. Communications in Numerical Methods in Engineering, 17, 893-902.
http://dx.doi.org/10.1002/cnm.452 [10] Puckett, G.P., Almgren, A.S., Bell, J.B., Marcus, D.L. and Rider, W. (1997) A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows. Journal of Computational Physics, 130, 269-282.
http://dx.doi.org/10.1006/jcph.1996.5590 [11] Buscaglia, G.C. and Codina, R. (2000) Fourier Analysis of an Equal-Order Incompressible Flow Solver Stabilized by Pressure Gradient Projection. International Journal for Numerical Methods in Fluids, 34, 65-92.
http://dx.doi.org/10.1002/1097-0363(20000915)34:1<65::AID-FLD56>3.0.CO;2-J [12] Bell, J.B. and Marcus, D.L. (1992) A Second Order Projection Method for Variable-Density Flows. Journal of Computational Physics, 101, 334-348. http://dx.doi.org/10.1016/0021-9991(92)90011-M [13] Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H. and Welcome, M.L. (1998) A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations. Journal of Computational Physics, 142, 1-46. http://dx.doi.org/10.1006/jcph.1998.5890 [14] Li, Y., Mei, L., Ge, J. and Shi, F. (2013) A New Fractional Time-Stepping Method for Variable Density Incompressible Flows. Journal of Computational Physics, 242, 124-137.
http://dx.doi.org/10.1016/j.jcp.2013.02.010 [15] Calgaro, C., Creuse, E. and Goudon, T. (2008) An Hybrid Finite Volume-Finite Element Method for Variable Density Incompressible Flows. Journal of Computational Physics, 227, 4671-4696.
http://dx.doi.org/10.1016/j.jcp.2008.01.017 [16] Guermond, J.L. (1999) Stabilization of Galerkin Approximations of Transport Equations by Subgrid Modeling. ESAIM: Mathematical Modelling and Numerical Analysis, 33, 1293-1316.
http://dx.doi.org/10.1051/m2an:1999145 [17] Pyo, J. and Shen, J. (2007) Gauge-Uzawa Methods for Incompressible Flows with Variable Density. Journal of Computational Physics, 221, 181-197.
http://dx.doi.org/10.1016/j.jcp.2006.06.013 [18] Boukir, K. and Maday, Y. (1997) A High-Order Characteristics/Finite Element Method for the Incompressible Navier-Stokes Equations. International Journal for Numerical Methods in Fluids, 25, 1421-1454.
http://dx.doi.org/10.1002/(SICI)1097-0363(19971230)25:12<1421::AID-FLD334>3.0.CO;2-A [19] Douglas, J., Thomas, F. and Russell, T. (1982) Numerical Method for Convection-Dominated Diffusion Problem Based on Combining the Method of Characteristics with Finite Element of Finite Difference Procedures. SIAM Journal on Numerical Analysis, 19, 871-885.
http://dx.doi.org/10.1137/0719063 [20] Pironneau, O. (1982) On the Transport-Diffusion Algorithm and Its Application to the Navier-Stokes Equations. Numerische Mathematik, 38, 309-332.
http://dx.doi.org/10.1007/BF01396435 [21] Süli, E. (1998) Convergence and Nonlinear Stability of the Lagrange-Galerkin Method for the Navier-Stokes Equations. Numerische Mathematik, 53, 459-483. [22] Temam, R. (2001) Navier-Stokes Equations, Theory and Numerical Analysis. Reprint of the 1984 Edition, AMS Chelsea Publishing, Providence. [23] Girault, V. and Raviart, P.A. (1986) Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-61623-5 [24] Ern, A. and Guermond, J.L. (2004) Theory and Practice of Finite Elements. Applied Mathematical Sciences, Vol. 159, Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4757-4355-5 [25] Adams, R.A. (1975) Sobolev Spaces. Academic Press, New York. [26] Evans, L.C. (1998) Partial Differential Equations. American Mathematical Society, Providence. [27] Chen, Z. (2005) Finite Element Methods and Their Applications (Scientific Computation). Springer, Berlin. [28] Benítez, M. and Bermúdez, A. (2011) A Second Order Characteristics Finite Element Scheme for Natural Convection Problems. Journal of Computational and Applied Mathematics, 235, 3270-3284.
http://dx.doi.org/10.1016/j.cam.2011.01.007 [29] Arbogast, T. and Wheeler, M.F. (1995) A Characteristics-Mixed Finite Element Method for Advection-Dominated Transport-Problems. SIAM Journal on Numerical Analysis, 32, 404-424.
http://dx.doi.org/10.1137/0732017 [30] Turek, S. (1999) Efficient Solvers for Incompressible Flow Problems. Lecture Notes in Computer Science, Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-58393-3 [31] Codina, R., Blasco, J., Buscaglia, G. and Huerta, A. (2001) Implementation of a Stabilized Finite Element Formulation for the Incompressible Navier-Stokes Equations Based on a Pressure Gradient Projection. International Journal for Numerical Methods in Fluids, 37, 419-444.
http://dx.doi.org/10.1002/fld.182 [32] Dohrmann, C. and Bochev, P. (2004) A Stabilized Finite Element Method for the Stokes Problem Based on Polynomial Pressure Projections. International Journal for Numerical Methods in Fluids, 46, 183-201.
http://dx.doi.org/10.1002/fld.752 [33] He, Y. and Li, J. (2009) Convergence of Three Iterative Methods Based on the Finite Element Discretization for the Stationary Navier-Stokes Equations. Computer Methods in Applied Mechanics and Engineering, 198, 1351-1359.
http://dx.doi.org/10.1016/j.cma.2008.12.001 [34] Guermond, J.L. and Salgado, A.J. (2011) Error Analysis of a Fractional Time-Stepping Technique for Incompressible Flows with Variable Density. SIAM Journal on Numerical Analysis, 49, 917-944.
http://dx.doi.org/10.1137/090768758 [35] Tryggvason, G. (1988) Numerical Simulations of the Rayleigh-Taylor Instability. Journal of Computational Physics, 75, 253-282.
http://dx.doi.org/10.1016/0021-9991(88)90112-X [36] Schneider, T., Botta, N., Geratz, K.J. and Klein, R. (1999) Extension of Finite Volume Compressible Flow Solvers to Multidimensional Variable Density Zero Mach Number Flows. Journal of Computational Physics, 155, 248-286.
http://dx.doi.org/10.1006/jcph.1999.6327 [37] Rasthofer, U., Henke, F., Wall, W.A. and Gravemeier, V. (2011) An Extended Residual-Based Variational Multiscale Method for Two-Phase Flow Including Surface Tension. Computer Methods in Applied Mechanics and Engineering, 200, 1866-1876.
http://dx.doi.org/10.1016/j.cma.2011.02.004 [38] Fries, T.P. (2010) Level Set Method for Two-Phase Incompressible Flows under Magnetic Fields. Computer Physics Communications, 181, 999-1007.
http://dx.doi.org/10.1016/j.cpc.2010.02.002 [39] Fries, T.P. (2009) The Intrinsic XFEM for Two-Phase Flows. International Journal for Numerical Methods in Fluids, 60, 437-471.
http://dx.doi.org/10.1002/fld.1901
[1] Guermond, J.L. and Salgado, A. (2009) A Splitting Method for Incompressible Flows with Variable Density Based on a Pressure Poisson Equation. Journal of Computational Physics, 228, 2834-2846.
http://dx.doi.org/10.1016/j.jcp.2008.12.036 [2] Pyo, J.H. and Shen, J. (2007) Gauge-Uzawa Methods for Incompressible Flows with Variable Density. Journal of Computational Physics, 221, 181-197.
http://dx.doi.org/10.1016/j.jcp.2006.06.013 [3] Guermond, J.L. and Quartapelle, L. (2000) A Projection FEM for Variable Density Incompressible Flows. Journal of Computational Physics, 165, 167-188.
http://dx.doi.org/10.1006/jcph.2000.6609 [4] Chorin, A.J. (1968) Numeriacl Solution of the Navier-Stokes Equations. Mathematics of Computation, 22, 745-762.
http://dx.doi.org/10.1090/S0025-5718-1968-0242392-2 [5] Chorin, A.J. (1969) On the Convergence of Discrete Approximations to the Navier-Stokes Equations. Mathematics of Computation, 23, 341-353. http://dx.doi.org/10.1090/S0025-5718-1969-0242393-5 [6] Temam, R. (1977) Navier-Stokes Equations, Studies in Mathematics and Its Applications 2. North-Holland, Amsterdam. [7] Temam, R. (1969) Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II). Archive for Rational Mechanics and Analysis, 33, 377-385.
http://dx.doi.org/10.1007/BF00247696 [8] Guermond, J.L., Minev, P. and Shen, J. (2006) An Overview of Projection Methods for Incompressible Flows. Computer Methods in Applied Mechanics and Engineering, 195, 6011-6045.
http://dx.doi.org/10.1016/j.cma.2005.10.010 [9] Fraigneau, Y., Guermond, J.L. and Quartapelle, L. (2001) Approximation of Variable Density Incompressible Flows by Means of Finite Elements and Finite Volumes. Communications in Numerical Methods in Engineering, 17, 893-902.
http://dx.doi.org/10.1002/cnm.452 [10] Puckett, G.P., Almgren, A.S., Bell, J.B., Marcus, D.L. and Rider, W. (1997) A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows. Journal of Computational Physics, 130, 269-282.
http://dx.doi.org/10.1006/jcph.1996.5590 [11] Buscaglia, G.C. and Codina, R. (2000) Fourier Analysis of an Equal-Order Incompressible Flow Solver Stabilized by Pressure Gradient Projection. International Journal for Numerical Methods in Fluids, 34, 65-92.
http://dx.doi.org/10.1002/1097-0363(20000915)34:1<65::AID-FLD56>3.0.CO;2-J [12] Bell, J.B. and Marcus, D.L. (1992) A Second Order Projection Method for Variable-Density Flows. Journal of Computational Physics, 101, 334-348. http://dx.doi.org/10.1016/0021-9991(92)90011-M [13] Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H. and Welcome, M.L. (1998) A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations. Journal of Computational Physics, 142, 1-46. http://dx.doi.org/10.1006/jcph.1998.5890 [14] Li, Y., Mei, L., Ge, J. and Shi, F. (2013) A New Fractional Time-Stepping Method for Variable Density Incompressible Flows. Journal of Computational Physics, 242, 124-137.
http://dx.doi.org/10.1016/j.jcp.2013.02.010 [15] Calgaro, C., Creuse, E. and Goudon, T. (2008) An Hybrid Finite Volume-Finite Element Method for Variable Density Incompressible Flows. Journal of Computational Physics, 227, 4671-4696.
http://dx.doi.org/10.1016/j.jcp.2008.01.017 [16] Guermond, J.L. (1999) Stabilization of Galerkin Approximations of Transport Equations by Subgrid Modeling. ESAIM: Mathematical Modelling and Numerical Analysis, 33, 1293-1316.
http://dx.doi.org/10.1051/m2an:1999145 [17] Pyo, J. and Shen, J. (2007) Gauge-Uzawa Methods for Incompressible Flows with Variable Density. Journal of Computational Physics, 221, 181-197.
http://dx.doi.org/10.1016/j.jcp.2006.06.013 [18] Boukir, K. and Maday, Y. (1997) A High-Order Characteristics/Finite Element Method for the Incompressible Navier-Stokes Equations. International Journal for Numerical Methods in Fluids, 25, 1421-1454.
http://dx.doi.org/10.1002/(SICI)1097-0363(19971230)25:12<1421::AID-FLD334>3.0.CO;2-A [19] Douglas, J., Thomas, F. and Russell, T. (1982) Numerical Method for Convection-Dominated Diffusion Problem Based on Combining the Method of Characteristics with Finite Element of Finite Difference Procedures. SIAM Journal on Numerical Analysis, 19, 871-885.
http://dx.doi.org/10.1137/0719063 [20] Pironneau, O. (1982) On the Transport-Diffusion Algorithm and Its Application to the Navier-Stokes Equations. Numerische Mathematik, 38, 309-332.
http://dx.doi.org/10.1007/BF01396435 [21] Süli, E. (1998) Convergence and Nonlinear Stability of the Lagrange-Galerkin Method for the Navier-Stokes Equations. Numerische Mathematik, 53, 459-483. [22] Temam, R. (2001) Navier-Stokes Equations, Theory and Numerical Analysis. Reprint of the 1984 Edition, AMS Chelsea Publishing, Providence. [23] Girault, V. and Raviart, P.A. (1986) Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-61623-5 [24] Ern, A. and Guermond, J.L. (2004) Theory and Practice of Finite Elements. Applied Mathematical Sciences, Vol. 159, Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4757-4355-5 [25] Adams, R.A. (1975) Sobolev Spaces. Academic Press, New York. [26] Evans, L.C. (1998) Partial Differential Equations. American Mathematical Society, Providence. [27] Chen, Z. (2005) Finite Element Methods and Their Applications (Scientific Computation). Springer, Berlin. [28] Benítez, M. and Bermúdez, A. (2011) A Second Order Characteristics Finite Element Scheme for Natural Convection Problems. Journal of Computational and Applied Mathematics, 235, 3270-3284.
http://dx.doi.org/10.1016/j.cam.2011.01.007 [29] Arbogast, T. and Wheeler, M.F. (1995) A Characteristics-Mixed Finite Element Method for Advection-Dominated Transport-Problems. SIAM Journal on Numerical Analysis, 32, 404-424.
http://dx.doi.org/10.1137/0732017 [30] Turek, S. (1999) Efficient Solvers for Incompressible Flow Problems. Lecture Notes in Computer Science, Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-58393-3 [31] Codina, R., Blasco, J., Buscaglia, G. and Huerta, A. (2001) Implementation of a Stabilized Finite Element Formulation for the Incompressible Navier-Stokes Equations Based on a Pressure Gradient Projection. International Journal for Numerical Methods in Fluids, 37, 419-444.
http://dx.doi.org/10.1002/fld.182 [32] Dohrmann, C. and Bochev, P. (2004) A Stabilized Finite Element Method for the Stokes Problem Based on Polynomial Pressure Projections. International Journal for Numerical Methods in Fluids, 46, 183-201.
http://dx.doi.org/10.1002/fld.752 [33] He, Y. and Li, J. (2009) Convergence of Three Iterative Methods Based on the Finite Element Discretization for the Stationary Navier-Stokes Equations. Computer Methods in Applied Mechanics and Engineering, 198, 1351-1359.
http://dx.doi.org/10.1016/j.cma.2008.12.001 [34] Guermond, J.L. and Salgado, A.J. (2011) Error Analysis of a Fractional Time-Stepping Technique for Incompressible Flows with Variable Density. SIAM Journal on Numerical Analysis, 49, 917-944.
http://dx.doi.org/10.1137/090768758 [35] Tryggvason, G. (1988) Numerical Simulations of the Rayleigh-Taylor Instability. Journal of Computational Physics, 75, 253-282.
http://dx.doi.org/10.1016/0021-9991(88)90112-X [36] Schneider, T., Botta, N., Geratz, K.J. and Klein, R. (1999) Extension of Finite Volume Compressible Flow Solvers to Multidimensional Variable Density Zero Mach Number Flows. Journal of Computational Physics, 155, 248-286.
http://dx.doi.org/10.1006/jcph.1999.6327 [37] Rasthofer, U., Henke, F., Wall, W.A. and Gravemeier, V. (2011) An Extended Residual-Based Variational Multiscale Method for Two-Phase Flow Including Surface Tension. Computer Methods in Applied Mechanics and Engineering, 200, 1866-1876.
http://dx.doi.org/10.1016/j.cma.2011.02.004 [38] Fries, T.P. (2010) Level Set Method for Two-Phase Incompressible Flows under Magnetic Fields. Computer Physics Communications, 181, 999-1007.
http://dx.doi.org/10.1016/j.cpc.2010.02.002 [39] Fries, T.P. (2009) The Intrinsic XFEM for Two-Phase Flows. International Journal for Numerical Methods in Fluids, 60, 437-471.
http://dx.doi.org/10.1002/fld.1901