# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155
##### References:
 [1] J. Adams, P. Swarztrauber and R. Sweet, FISHPACK: Efficient FORTRAN subprograms for the solution of separable elliptic partial differential equations. Available from: http://www.netlib.org/fishpack/. [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), 257-283. doi: 10.1016/0021-9991(89)90151-4. [3] D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168 (2001), 464-499. doi: 10.1006/jcph.2001.6715. [4] D. Calhoun, A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., 176 (2002), 231-275. doi: 10.1006/jcph.2001.6970. [5] D. L. Chopp, Some improvements of the fast marching method, SIAM J. Sci. Comput., 23 (2001), 230-244. doi: 10.1137/S106482750037617X. [6] S. Deng, "Immersed Interface Method for Three Dimensional Interface Problems and Applications," Ph.D thesis, North Carolina State University, 2001. [7] S. Deng, K. Ito and Z. Li, Three-dimensional elliptic solvers for interface problems and applications, J. Comput. Phys., 184 (2003), 215-243. doi: 10.1016/S0021-9991(02)00028-1. [8] W. E and J.-G. Liu, Projection method. I. Convergence and numerical boundary-layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047. [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), 236-252. doi: 10.1006/jcph.1997.5689. [10] J. Hunter, Z. Li and H. Zhao, Reactive autophobic spreading of drops, J. Comput. Phys., 183 (2002), 335-366. doi: 10.1006/jcph.2002.7168. [11] Z. Li, "The Immersed Interface Method: A Numerical Approach for Partial Differential Equations with Interfaces," Ph.D thesis, University of Washington, 1994. [12] Z. Li, A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35 (1998), 230-254. doi: 10.1137/S0036142995291329. [13] Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23 (2001), 339-361. doi: 10.1137/S1064827500370160. [14] Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDEs Involving Interfaces and Irregular Domains," Frontiers in Applied Mathematics, 33, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. [15] Z. Li, S. R. Lubkin and X. Wan, An augmented IIM-level set method for Stokes equations with discontinuous viscosity, in "Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations," Electron. J. Diff. Eqns. Conf., 15 Southwest Texas State Univ., San Marcos, TX, (2007), 193-210. [16] Z. Li and B. Soni, Fast and accurate numerical approaches for Stefan problems and crystal growth, Numerical Heat Transfer, B: Fundamentals, 35 (1999), 461-484. doi: 10.1080/104077999275848. [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), 281-304. doi: 10.1006/jcph.1999.6249. [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), 157-192. doi: 10.1016/j.jcp.2004.09.017. [19] S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003. [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), 177-205. doi: 10.1016/S0021-9991(03)00310-3. [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, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999. [22] X. Wan, "Numerical Simulation Methods for Biological Tissue Interactions," Ph.D thesis, North Carolina State University, 2007. [23] W.-J. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phy., 227 (2007), 1046-1074. doi: 10.1016/j.jcp.2007.08.021. [24] P. M. de Zeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, J. Comput. Appl. Math., 33 (1990), 1-27. doi: 10.1016/0377-0427(90)90252-U.

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##### References:
 [1] J. Adams, P. Swarztrauber and R. Sweet, FISHPACK: Efficient FORTRAN subprograms for the solution of separable elliptic partial differential equations. Available from: http://www.netlib.org/fishpack/. [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), 257-283. doi: 10.1016/0021-9991(89)90151-4. [3] D. L. Brown, R. Cortez and M. L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168 (2001), 464-499. doi: 10.1006/jcph.2001.6715. [4] D. Calhoun, A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., 176 (2002), 231-275. doi: 10.1006/jcph.2001.6970. [5] D. L. Chopp, Some improvements of the fast marching method, SIAM J. Sci. Comput., 23 (2001), 230-244. doi: 10.1137/S106482750037617X. [6] S. Deng, "Immersed Interface Method for Three Dimensional Interface Problems and Applications," Ph.D thesis, North Carolina State University, 2001. [7] S. Deng, K. Ito and Z. Li, Three-dimensional elliptic solvers for interface problems and applications, J. Comput. Phys., 184 (2003), 215-243. doi: 10.1016/S0021-9991(02)00028-1. [8] W. E and J.-G. Liu, Projection method. I. Convergence and numerical boundary-layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047. [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), 236-252. doi: 10.1006/jcph.1997.5689. [10] J. Hunter, Z. Li and H. Zhao, Reactive autophobic spreading of drops, J. Comput. Phys., 183 (2002), 335-366. doi: 10.1006/jcph.2002.7168. [11] Z. Li, "The Immersed Interface Method: A Numerical Approach for Partial Differential Equations with Interfaces," Ph.D thesis, University of Washington, 1994. [12] Z. Li, A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35 (1998), 230-254. doi: 10.1137/S0036142995291329. [13] Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23 (2001), 339-361. doi: 10.1137/S1064827500370160. [14] Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDEs Involving Interfaces and Irregular Domains," Frontiers in Applied Mathematics, 33, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. [15] Z. Li, S. R. Lubkin and X. Wan, An augmented IIM-level set method for Stokes equations with discontinuous viscosity, in "Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations," Electron. J. Diff. Eqns. Conf., 15 Southwest Texas State Univ., San Marcos, TX, (2007), 193-210. [16] Z. Li and B. Soni, Fast and accurate numerical approaches for Stefan problems and crystal growth, Numerical Heat Transfer, B: Fundamentals, 35 (1999), 461-484. doi: 10.1080/104077999275848. [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), 281-304. doi: 10.1006/jcph.1999.6249. [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), 157-192. doi: 10.1016/j.jcp.2004.09.017. [19] S. Osher and R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003. [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), 177-205. doi: 10.1016/S0021-9991(03)00310-3. [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, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999. [22] X. Wan, "Numerical Simulation Methods for Biological Tissue Interactions," Ph.D thesis, North Carolina State University, 2007. [23] W.-J. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phy., 227 (2007), 1046-1074. doi: 10.1016/j.jcp.2007.08.021. [24] P. M. de Zeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, J. Comput. Appl. Math., 33 (1990), 1-27. doi: 10.1016/0377-0427(90)90252-U.
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