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A Novel Decomposition as a Fast Finite Difference Method for Second Derivatives

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Abstract

In this paper, we propose a fast solver for the linear system-induced from applying the finite difference method. For this purpose, we provide a new decomposition of the matrix consisting of a second-order central finite difference matrix and a small condition number matrix. We analyze the fourth-order finite difference matrix using this decomposition and the good properties of the two decomposed matrices. In addition, we compare the upper bounds of the condition numbers of the three matrices, which are closely related to the number of iterations. In terms of computational cost, we show the superiority of the proposed solver by solving the one-dimensional Poisson’s equations. We also demonstrated the efficiency of using the proposed solver by numerically observing the factors, such as condition number and spectral radius.

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Funding

The author Bak was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01004139, No. 2022R1I1A1A01059167). The author Park was supported by the R &D program of through Korea Institute of Fusion Energy (KFE) funded by the Government funds, Republic of Korea (No. EN2241-3) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C1A01008506, No. 2022R1I1A1A01053950).

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  1. Yonghyeon Jeon and Sangbeom Park contributed equally to this study.

Authors and Affiliations

  1. Department of Mathematics, Kyungpook National University, 80 Daehak-ro, Daegu, 41566, Republic of Korea

    Soyoon Bak, Yonghyeon Jeon & Sangbeom Park

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  1. Soyoon Bak
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  2. Yonghyeon Jeon
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  3. Sangbeom Park
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Correspondence to Soyoon Bak.

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Bak, S., Jeon, Y. & Park, S. A Novel Decomposition as a Fast Finite Difference Method for Second Derivatives. Results Math 78, 22 (2023). https://doi.org/10.1007/s00025-022-01798-y

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