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Volume 11, Issue 4
Pollution-Free Finite Difference Schemes for Non-Homogeneous Helmholtz Equation

K. Wang & Y. S. Wong

Int. J. Numer. Anal. Mod., 11 (2014), pp. 787-815.

Published online: 2014-11

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  • Abstract

In this paper, we develop pollution-free finite difference schemes for solving the non-homogeneous Helmholtz equation in one dimension. A family of high-order algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features of the proposed schemes are: first, the new difference schemes have a $2n$-order of rate of convergence and are pollution-free. Hence, the error is bounded even for the equation at high wave numbers. Secondly, the resulting difference scheme is simple, namely it has the same structure as the standard three-point central differencing regardless of the order of accuracy. Convergence analysis is presented, and numerical simulations are reported for the non-homogeneous Helmholtz equation with both constant and varying wave numbers. The computational results clearly confirm the superior performance of the proposed schemes.

  • AMS Subject Headings

65N06, 65N15, 65N22

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-787, author = {Wang , K. and Wong , Y. S.}, title = {Pollution-Free Finite Difference Schemes for Non-Homogeneous Helmholtz Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {4}, pages = {787--815}, abstract = {

In this paper, we develop pollution-free finite difference schemes for solving the non-homogeneous Helmholtz equation in one dimension. A family of high-order algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features of the proposed schemes are: first, the new difference schemes have a $2n$-order of rate of convergence and are pollution-free. Hence, the error is bounded even for the equation at high wave numbers. Secondly, the resulting difference scheme is simple, namely it has the same structure as the standard three-point central differencing regardless of the order of accuracy. Convergence analysis is presented, and numerical simulations are reported for the non-homogeneous Helmholtz equation with both constant and varying wave numbers. The computational results clearly confirm the superior performance of the proposed schemes.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/552.html} }
TY - JOUR T1 - Pollution-Free Finite Difference Schemes for Non-Homogeneous Helmholtz Equation AU - Wang , K. AU - Wong , Y. S. JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 787 EP - 815 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/552.html KW - Helmholtz equation, Finite difference method, Convergence analysis, High wave number, Pollution-free, High-order schemes. AB -

In this paper, we develop pollution-free finite difference schemes for solving the non-homogeneous Helmholtz equation in one dimension. A family of high-order algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features of the proposed schemes are: first, the new difference schemes have a $2n$-order of rate of convergence and are pollution-free. Hence, the error is bounded even for the equation at high wave numbers. Secondly, the resulting difference scheme is simple, namely it has the same structure as the standard three-point central differencing regardless of the order of accuracy. Convergence analysis is presented, and numerical simulations are reported for the non-homogeneous Helmholtz equation with both constant and varying wave numbers. The computational results clearly confirm the superior performance of the proposed schemes.

K. Wang & Y. S. Wong. (1970). Pollution-Free Finite Difference Schemes for Non-Homogeneous Helmholtz Equation. International Journal of Numerical Analysis and Modeling. 11 (4). 787-815. doi:
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