The authors construct a new class of backward difference formula (BDF) and implicit-explicit (IMEX) schemes for solving parabolic type equations. The key idea is to base the schemes on Taylor expansions at time tn+β, where β > 1 is a tunable parameter, rather than the classical case of β = 1.
The main highlights and insights are:
The new schemes generalize the classical BDF and IMEX schemes, with the same computational effort but improved stability properties. By choosing a suitable β > 1, the stability regions of the higher-order schemes can be significantly enlarged compared to the classical case.
The authors identify an explicit and uniform multiplier for the new class of BDF and IMEX schemes up to fourth-order, which is crucial for establishing rigorous stability and error analysis using energy arguments.
For linear parabolic equations, the new schemes are shown to be unconditionally stable. For nonlinear parabolic equations, the stability and error analysis demonstrate that the new schemes become less restrictive as β increases, especially compared to the classical case of β = 1.
Numerical examples are provided to validate the theoretical findings, showing the advantages of the new schemes in terms of allowing larger time steps at higher-order accuracy.
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by Fukeng Huang... lúc arxiv.org 05-02-2024
https://arxiv.org/pdf/2405.00300.pdfYêu cầu sâu hơn