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Volume 16, Issue 2
New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Kejia Pan, Kang Fu, Jin Li, Hongling Hu & Zhilin Li

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 393-409.

Published online: 2023-04

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  • Abstract

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

  • AMS Subject Headings

65N06, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-393, author = {Pan , KejiaFu , KangLi , JinHu , Hongling and Li , Zhilin}, title = {New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {2}, pages = {393--409}, abstract = {

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0073}, url = {http://global-sci.org/intro/article_detail/nmtma/21582.html} }
TY - JOUR T1 - New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations AU - Pan , Kejia AU - Fu , Kang AU - Li , Jin AU - Hu , Hongling AU - Li , Zhilin JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 393 EP - 409 PY - 2023 DA - 2023/04 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0073 UR - https://global-sci.org/intro/article_detail/nmtma/21582.html KW - Poisson equation, Helmholtz equation, sixth-order compact scheme, maximum principle, staggered grid. AB -

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

Kejia Pan, Kang Fu, Jin Li, Hongling Hu & Zhilin Li. (2023). New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations. Numerical Mathematics: Theory, Methods and Applications. 16 (2). 393-409. doi:10.4208/nmtma.OA-2022-0073
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