Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T10:40:34.333Z Has data issue: false hasContentIssue false
  • English
  • Français

A Second Order Ghost Fluid Method for an Interface Problem of the Poisson Equation

Part of: Basic methods in fluid mechanics

Published online by Cambridge University Press:  28 July 2017

Cheng Liu  and
Changhong Hu
Cheng Liu*
Affiliation:
Research Institute for Applied Mechanics, Kyushu University, Fukuoka 816-0811, Japan
Changhong Hu*
Affiliation:
Research Institute for Applied Mechanics, Kyushu University, Fukuoka 816-0811, Japan
*
*Corresponding author. Email addresses:liu@riam.kyushu-u.ac.jp (C. Liu), hu@riam.kyushu-u.ac.jp (C. Hu)
*Corresponding author. Email addresses:liu@riam.kyushu-u.ac.jp (C. Liu), hu@riam.kyushu-u.ac.jp (C. Hu)
Get access

Abstract

A second order Ghost Fluid method is proposed for the treatment of interface problems of elliptic equations with discontinuous coefficients. By appropriate use of auxiliary virtual points, physical jump conditions are enforced at the interface. The signed distance function is used for the implicit description of irregular domain. With the additional unknowns, high order approximation considering the discontinuity can be built. To avoid the ill-conditioned matrix, the interpolation stencils are selected adaptively to balance the accuracy and the numerical stability. Additional equations containing the jump restrictions are assembled with the original discretized algebraic equations to form a new sparse linear system. Several Krylov iterative solvers are tested for the newly derived linear system. The results of a series of 1-D, 2-D tests show that the proposed method possesses second order accuracy in L∞ norm. Besides, the method can be extended to the 3-D problems straightforwardly. Numerical results reveal the present method is highly efficient and robust in dealing with the interface problems of elliptic equations.

Keywords

Interface problemghost fluid methodPoisson equationjump conditions

MSC classification

Secondary: 65Z05: Applications to physics 76M20: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 4 , October 2017 , pp. 965 - 996
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ashton, S. R. and Alexei, F. C., A matlab-based finite difference solver for the Poisson problem with mixed Dirichlet-Neumann boundary conditions, Comput. Phys. Commun., 184 (2012), 783–798.Google Scholar
[2] Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175–202.CrossRefGoogle Scholar
[3] Peskin, C. S., Flow patterns around heart valves: A numerical method, J. Comput. Phys., 10 (1972), 252–271.CrossRefGoogle Scholar
[4] Mohd-Yusof, J., Combined immersed-boundary/B-spline methods for simulations of flow in complex geometries, Center for Turbulence Research Annual Research Briefs, 161 (1997), 317–327.Google Scholar
[5] LeVeque, R. J. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019–1044.Google Scholar
[6] Li, Z. and Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23 (2001), 339–361.Google Scholar
[7] Zhong, X., A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, J. Comput. Phys., 225 (2007), 1066–1099.Google Scholar
[8] Fedkiw, R., Aslam, T., Merriman, B. and Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), 457.Google Scholar
[9] Liu, X. D., Fedkiw, R. P. and Kang, M., A boundary condition capturing method for Poissons equation on irregular domains, J. Comput. Phys., 160 (2000), 151–178.Google Scholar
[10] Zhou, Y., Zhao, S., Feig, M. and Wei, G., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., 213 (2006), 1–30.CrossRefGoogle Scholar
[11] Zhou, Y. and Wei, G., On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, J. Comput. Phys., 219 (2006), 228–246.Google Scholar
[12] Yu, S. and Wei, G., Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities, J. Comput. Phys., 227 (2007), 602–632.Google Scholar
[13] Johansen, H. and Colella, P., A Cartesian grid embedded boundary method for Poisson's equation on irregular domains, J. Comput. Phys., 147 (1998), 60–85.Google Scholar
[14] Cisternino, M. and Weynans, L., A parallel second order Cartesian method for elliptic interface problems, Tech. Rep., 7573 INRIA; 2011.Google Scholar
[15] Gibou, F., Fedkiw, R. P., Cheng, L. T. and Kang, M., A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176 (2002), 205–227.Google Scholar
[16] Bedrossian, J., Von Brecht, J. H. and Zhu, S. et al., A second order virtual node method for elliptic problems with interfaces and irregular domains, J. Comput. Phys., 229 (2010), 6405–6426.Google Scholar
[17] Hellrung, J. L., Wang, L. and Sifakis, E. et al., A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions, J. Comput. Phys., 231 (2012), 2015–2048.Google Scholar
[18] Choi, J., Oberoi, R., Edwards, J. and Rosati, J., An immersed boundary method for complex incompressible flows, J. Comput. Phys., 224 (2007), 757–784.Google Scholar
[19] Liu, C. and Hu, C. H., An efficient immersed boundary treatment for complex moving object, J. Comput. Phys., 274 (2014), 654–680.Google Scholar
[20] Saad, Y., Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996.Google Scholar