June  2012, 17(4): 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces

1. 

Lilly Corporate Center, DC 4108, Eli Lilly and Company, Indiana, IN 46285, United States

2. 

Center For Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205

Received  December 2010 Revised  September 2011 Published  February 2012

Solving a Helmholtz equation $\Delta u + \lambda u = f$ efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of $\lambda$ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient $\lambda$ is inversely proportional to the mesh size.
Citation: Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155
References:
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S. Deng, "Immersed Interface Method for Three Dimensional Interface Problems and Applications,", Ph.D thesis, (2001).   Google Scholar

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S. Deng, K. Ito and Z. Li, Three-dimensional elliptic solvers for interface problems and applications,, J. Comput. Phys., 184 (2003), 215.  doi: 10.1016/S0021-9991(02)00028-1.  Google Scholar

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W. E and J.-G. Liu, Projection method. I. Convergence and numerical boundary-layers,, SIAM J. Numer. Anal., 32 (1995), 1017.  doi: 10.1137/0732047.  Google Scholar

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T. Hou, Z. Li, S. Osher and H. Zhao, A hybrid method for moving interface problems with application to the Hele-Shaw flow,, J. Comput. Phys., 134 (1997), 236.  doi: 10.1006/jcph.1997.5689.  Google Scholar

[10]

J. Hunter, Z. Li and H. Zhao, Reactive autophobic spreading of drops,, J. Comput. Phys., 183 (2002), 335.  doi: 10.1006/jcph.2002.7168.  Google Scholar

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Z. Li, "The Immersed Interface Method: A Numerical Approach for Partial Differential Equations with Interfaces,", Ph.D thesis, (1994).   Google Scholar

[12]

Z. Li, A fast iterative algorithm for elliptic interface problems,, SIAM J. Numer. Anal., 35 (1998), 230.  doi: 10.1137/S0036142995291329.  Google Scholar

[13]

Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients,, SIAM J. Sci. Comput., 23 (2001), 339.  doi: 10.1137/S1064827500370160.  Google Scholar

[14]

Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDEs Involving Interfaces and Irregular Domains,", Frontiers in Applied Mathematics, 33 (2006).   Google Scholar

[15]

Z. Li, S. R. Lubkin and X. Wan, An augmented IIM-level set method for Stokes equations with discontinuous viscosity,, in, 15 (2007), 193.   Google Scholar

[16]

Z. Li and B. Soni, Fast and accurate numerical approaches for Stefan problems and crystal growth,, Numerical Heat Transfer, 35 (1999), 461.  doi: 10.1080/104077999275848.  Google Scholar

[17]

Z. Li, H. Zhao and H. Gao, A numerical study of electro-migration voiding by evolving level set functions on a fixed cartesian grid,, J. Comput. Phys., 152 (1999), 281.  doi: 10.1006/jcph.1999.6249.  Google Scholar

[18]

M. N. Linnick and H. F. Fasel, A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains,, J. Comput. Phys., 204 (2005), 157.  doi: 10.1016/j.jcp.2004.09.017.  Google Scholar

[19]

S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces,", Applied Mathematical Sciences, 153 (2003).   Google Scholar

[20]

D. Russell and Z. J. Wang, A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow,, J. Comput. Phys., 191 (2003), 177.  doi: 10.1016/S0021-9991(03)00310-3.  Google Scholar

[21]

J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science,", 2nd edition, 3 (1999).   Google Scholar

[22]

X. Wan, "Numerical Simulation Methods for Biological Tissue Interactions,", Ph.D thesis, (2007).   Google Scholar

[23]

W.-J. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems,, J. Comput. Phy., 227 (2007), 1046.  doi: 10.1016/j.jcp.2007.08.021.  Google Scholar

[24]

P. M. de Zeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver,, J. Comput. Appl. Math., 33 (1990), 1.  doi: 10.1016/0377-0427(90)90252-U.  Google Scholar

show all references

References:
[1]

J. Adams, P. Swarztrauber and R. Sweet, FISHPACK: Efficient FORTRAN subprograms for the solution of separable elliptic partial differential equations., Available from: \url{http://www.netlib.org/fishpack/}., ().   Google Scholar

[2]

J. B. Bell, P. Colella and H. M. Glaz., A second-order projection method for the incompressible Navier-Stokes equations,, J. Comput. Phys., 85 (1989), 257.  doi: 10.1016/0021-9991(89)90151-4.  Google Scholar

[3]

D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations,, J. Comput. Phys., 168 (2001), 464.  doi: 10.1006/jcph.2001.6715.  Google Scholar

[4]

D. Calhoun, A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions,, J. Comput. Phys., 176 (2002), 231.  doi: 10.1006/jcph.2001.6970.  Google Scholar

[5]

D. L. Chopp, Some improvements of the fast marching method,, SIAM J. Sci. Comput., 23 (2001), 230.  doi: 10.1137/S106482750037617X.  Google Scholar

[6]

S. Deng, "Immersed Interface Method for Three Dimensional Interface Problems and Applications,", Ph.D thesis, (2001).   Google Scholar

[7]

S. Deng, K. Ito and Z. Li, Three-dimensional elliptic solvers for interface problems and applications,, J. Comput. Phys., 184 (2003), 215.  doi: 10.1016/S0021-9991(02)00028-1.  Google Scholar

[8]

W. E and J.-G. Liu, Projection method. I. Convergence and numerical boundary-layers,, SIAM J. Numer. Anal., 32 (1995), 1017.  doi: 10.1137/0732047.  Google Scholar

[9]

T. Hou, Z. Li, S. Osher and H. Zhao, A hybrid method for moving interface problems with application to the Hele-Shaw flow,, J. Comput. Phys., 134 (1997), 236.  doi: 10.1006/jcph.1997.5689.  Google Scholar

[10]

J. Hunter, Z. Li and H. Zhao, Reactive autophobic spreading of drops,, J. Comput. Phys., 183 (2002), 335.  doi: 10.1006/jcph.2002.7168.  Google Scholar

[11]

Z. Li, "The Immersed Interface Method: A Numerical Approach for Partial Differential Equations with Interfaces,", Ph.D thesis, (1994).   Google Scholar

[12]

Z. Li, A fast iterative algorithm for elliptic interface problems,, SIAM J. Numer. Anal., 35 (1998), 230.  doi: 10.1137/S0036142995291329.  Google Scholar

[13]

Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients,, SIAM J. Sci. Comput., 23 (2001), 339.  doi: 10.1137/S1064827500370160.  Google Scholar

[14]

Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDEs Involving Interfaces and Irregular Domains,", Frontiers in Applied Mathematics, 33 (2006).   Google Scholar

[15]

Z. Li, S. R. Lubkin and X. Wan, An augmented IIM-level set method for Stokes equations with discontinuous viscosity,, in, 15 (2007), 193.   Google Scholar

[16]

Z. Li and B. Soni, Fast and accurate numerical approaches for Stefan problems and crystal growth,, Numerical Heat Transfer, 35 (1999), 461.  doi: 10.1080/104077999275848.  Google Scholar

[17]

Z. Li, H. Zhao and H. Gao, A numerical study of electro-migration voiding by evolving level set functions on a fixed cartesian grid,, J. Comput. Phys., 152 (1999), 281.  doi: 10.1006/jcph.1999.6249.  Google Scholar

[18]

M. N. Linnick and H. F. Fasel, A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains,, J. Comput. Phys., 204 (2005), 157.  doi: 10.1016/j.jcp.2004.09.017.  Google Scholar

[19]

S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces,", Applied Mathematical Sciences, 153 (2003).   Google Scholar

[20]

D. Russell and Z. J. Wang, A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow,, J. Comput. Phys., 191 (2003), 177.  doi: 10.1016/S0021-9991(03)00310-3.  Google Scholar

[21]

J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science,", 2nd edition, 3 (1999).   Google Scholar

[22]

X. Wan, "Numerical Simulation Methods for Biological Tissue Interactions,", Ph.D thesis, (2007).   Google Scholar

[23]

W.-J. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems,, J. Comput. Phy., 227 (2007), 1046.  doi: 10.1016/j.jcp.2007.08.021.  Google Scholar

[24]

P. M. de Zeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver,, J. Comput. Appl. Math., 33 (1990), 1.  doi: 10.1016/0377-0427(90)90252-U.  Google Scholar

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