Higher-order accurate finite volume discretization of Helmholtz equations with Pollution Effects Reductions

Yaw Kyei; Kossi Edoh

doi:10.31686/ijier.vol6.iss4.1017

Authors

Yaw Kyei

North Carolina Central University, USA
Kossi Edoh

North Carolina A & T State University, USA

DOI:

https://doi.org/10.31686/ijier.vol6.iss4.1017

Keywords:

Pollution Effects Reductions

Abstract

Higher-order accurate finite volume schemes are developed for Helmholtz equations in two dimensions. Through minimizations of local equation error expansions for the flux integral formulation of the equation, we determine quadrature weights for the discretization of the equation. Collocations of local expansions of the solution and the source terms are utilized to formulate weighted quadratures of all local compact fluxes to describe the equation error expansion within the computational domain. In using the source term distribution to account for fluxes along all compact directions about each grid point as the centroid of a local control volume, the right minimizing quadrature weights are determined and optimized for stability and uniform higher-order convergence. As a result, the resulting local residuals form more complete descriptions of the wave number k and the complexities of the associated pollution effects. The leading terms of the residual errors are optimized for pollution effects reductions to ensure stability and robust convergence of the resulting schemes. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

Author Biographies

Yaw Kyei, North Carolina Central University, USA

Department of Decision Sciences
Kossi Edoh, North Carolina A & T State University, USA

Department of Mathematics

References

[1] I. M. Babuska and Ss A. Sauter, Is the pollution effect of the fem avoidable for the helmholtz equation considering high wave numbers?, SIAM Journal on Numerical Analysis 34 (1997), no. 6, 2392–2423.
[2] G. Bao, G. W. Wei, and S. Zhao, Numerical solution of the helmholtz equation with high wavenumbers, Int. J. Numer. Meth. Engng 59 (2004), 389–408.
[3] Q. Du, J. R. Kamm, R. B. Lehoucq, and M. L. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM J. APPL. MATH. 72 (2012), no. 1, 464–487.
[4] H. C. Elman and D. P. OLeary, Efficient Iterative Solution of the Three-Dimensional Helmholtz Equation, J. Comput. Phys 142 (1998), 163–181.
[5] Y. Fu, Compact fourth-order finite difference schemes for helmholtz equation with high wave numbers, 26 (2008), 98–111.
[6] S. Gabersek and D. R. Durran, Gap Flows through Idealized Topography. Part II: Effects of Rotation and Surface Friction, J. Atmos. Sci. 63 (2006), 2720–2315.
[7] F. Ihlenburg and I. Babuska, Finite element solution of the Helmholtz equation with high wave number part i: the h-version of the FEM., Comp. Math. Appl. 30 (1995),
9–37.
[8] K. Ito, Y. Kyei, and Z. Li, Higher-Order, Cartesian Grid Based Finite Difference Schemes for Elliptic Equations on Irregular Domains , SIAM J. Sci. Comput. 27 (2005), 346–367.
[9] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, Inc, 2003.
[10] Y. Kyei, Space-time finite volume differencing framework for effective higher-order accurate discretizations of parabolic equations, SIAM J. Sci. Comput. 34 (2012), no. 3, A1432–A1459.
[11] Y. Kyei, J. P. Roop, and G. Tang, A family of sixth-order compact finite difference schemes for poisson equation, Adv. Numer. Anal. Article ID 352174 (2010), 1–17.
[12] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), 1–42.
[13] M. Li and T. Tang, A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows, J. Sci. Comput. 16 (2001), 29–45.
[14] F. Lorcher, G. Gassner, and C. D. Munz, An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations, J. Comput. Phys. 227 (2008), 5649–5670.
[15] J. Ma, J. Zhu, and M. Li, The galerkin boundary element method for exterior problems of 2-d helmholtz equation with arbitrary wavenumber, Engineering Analysis with Boundary Elements 34 (2010), no. 12, 1058 – 1063.
[16] W. H. Mason, Applied computational aerodynamics, Text/notes, http://www.dept.aoe.vt.edu/ maxon/Mason f/CAtextTop.html, 1997.
[17] O. Z. Mehdizadeh and M. Paraschivoiu, Investigation of a two-dimensional spectral element method for Helmholtz’s equation, J. Comput. Phys. 189 (2003), 111–129.
[18] P. Ming and X. Yue, Numerical methods for multiscale elliptic problems, J. Comput. Phys. 214 (2006), 421–445.
[19] J. T. Oden, S. Prudhomme, and L. Demkowicz, A posteriori error estimation for acoustic wave propagation problems, Archives of Computational Mechanics and Engineering 12 (2005), no. 4, 343–389.
[20] M. Piller and E. Stalio, Finite–volume compact schemes on staggered grids, J. Comput. Phys. 197 (2004), 1064–1094.
[21] J. Santos and P. de Oliveira, A converging finite volume scheme for hyperbolic conservation laws with source terms, J. Comput. App. Math. 111 (1999), 239–251.
[22] I. Singer and E. Turkel, High-order finite difference methods for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 163 (1998), 343–358.
[23] , Sixth Order Accurate Finite Difference Schemes for the Helmholtz Equation, J. Comput. Acoustics 14 (2006).
[24] W. F. Spotz and G. F. Carey, A high-order compact Scheme for the Steady StreamFunction Vorticity Equations, Int. J. Numer. Meth. Eng 38 (1995), 3497–3512.
[25] J. C. Strikwerda, Finite Difference and Partial Differential Equations, Wadsworth & BrooksCole Advanced Books & Software, 1989.
[26] H. Suna, N. Kanga, J. Zhang, and E. S. Carlson, A Fourth Order Compact Difference Scheme on Face Centered Cubic Grids with Multigrid Method for Solving 2D Convection Diffusion Equation, Mathematics and Computers in Simulation 63 (2003), 651–661.
[27] A. K. Verma, S. M. Bhallamudi, and V. Eswaran, Overlapping control volume method for solute transport, J. Hydr. Engrg. 5 (2000), 308–316.
[28] K. Wang and Y. S. Wong, Is Pollution Effect of Finite Difference Schemes Avoidable for Multi-Dimensional Helmholtz Equations with High Wave Numbers?, / Commun. Comput. Phys. 21, 490–514.
[29] E. Weinan and B. Engquist, Multiscale Modeling and Computation, Notices Of The AMS 50 (2003), no. 9, 1062–1070.
[30] Y. S. Wong and G. Li, Exact finite difference schemes for solving helmholtz equation at any wavenumber, Int. J. OF NUMER ANAL AND MODELING, SERIES B 2 (2011), no. 2, 91–108.
[31] A. Erlangga Y, C. Vuik, and C. W. Oosterlee, On A Robust Iterative Method For Heterogeneous Helmholtz Problems For Geophysics Applications, Int. J. Numer. Anal. Modeling 2 (2005), 197–208.