自然对流方程的一种新的二阶一致分裂格式
A new second-order consistent splitting scheme for the Natural Convection equations
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中文总结 AI 辅助
提出一种基于$t^{n+k}$泰勒展开的二阶一致分裂格式，证明$k>4$时稳定，以$k=5$为例，相比传统格式稳定性更优，并通过数值实验验证其精度与有效性。
中文摘要 AI 辅助
针对自然对流方程，提出了一种新的二阶一致分裂格式。该格式利用关于时间层$t^{n+k}$的泰勒展开构造，其中$k$为待定参数。证明了对于所有$k>4$该格式是稳定的，本研究主要聚焦于$k=5$的情形。与基于$t^{n+1}$处泰勒展开的传统分裂格式相比，所提出的格式展现出改进的稳定性。通过运用Sobolev不等式和Gronwall引理，我们严格建立了所提格式的稳定性，并在二维和三维空间维度下推导了误差估计。最后，通过若干数值实验验证了所提格式的稳定性和精度，并展示了其有效性。
英文摘要
A novel second-order consistent splitting scheme is proposed for the natural convection equations. The scheme is constructed using Taylor expansions about the time level $t^{n+k}$, where $k$ is a parameter to be determined. It is proved to be stable for all $k>4$, with the present study focusing primarily on the case $k=5$. Compared with the conventional splitting scheme based on Taylor expansions about $t^{n+1}$, the proposed scheme exhibits improved stability. By employing the Sobolev inequality and Gronwall's lemma, we rigorously establish stability of the proposed scheme and derive error estimates in both two and three spatial dimensions. Finally, several numerical experiments are presented to verify the stability and accuracy of the proposed scheme and to demonstrate its effectiveness.
发表机构
- Henan Polytechnic University（河南理工大学）
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