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Multigrid methods for time fractional conservation laws

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Abstract

For solving the time fractional conservation laws with discontinuous solutions such as shock waves, current feasible methods are the implicit finite-volume TVD schemes that employ the Lax-Friedrichs fluxes corrected by the limited slopes. However, the schemes are hard to implement when the fractional order \(\alpha \) is close to zero since in their implementation there are still no efficient methods to solve the strongly nonlinear spacial discrete systems at discrete times. We, aiming at increasing the implementation efficiency of the schemes above, develop two multigrid methods to solve these nonlinear spacial discrete systems. Numerical tests show that both multigrid methods have good convergence for relatively larger \(\alpha \), and the second one has an advantage in that its convergence is affected slightly when the order \(\alpha \) decreases to zero.

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Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11571053).

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  1. Huiran Bai contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics and Statistic, Changsha University of Science and Technology, Changsha, 410114, China

    Yingjun Jiang

  2. School of Mathematics and Statistic, Changsha University of Science and Technology, Changsha, 410114, China

    Huiran Bai

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  1. Yingjun Jiang
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  2. Huiran Bai
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Y. J. Jiang wrote the main manuscript and H. R. Bai did the numerical tests.

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Correspondence to Yingjun Jiang.

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Jiang, Y., Bai, H. Multigrid methods for time fractional conservation laws. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01750-x

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