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A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations

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Abstract

We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.

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Acknowledgements

The first author (D. Jeong) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2014R1A6A3A01009812). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2014R1A2A2A01003683). The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.

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  1. Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

    Darae Jeong & Junseok Kim

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  1. Darae Jeong
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  2. Junseok Kim
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Correspondence to Junseok Kim.

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Jeong, D., Kim, J. A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations. J Sci Comput 75, 332–349 (2018). https://doi.org/10.1007/s10915-017-0536-2

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