发表机构
Lund University（隆德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对通过低维界面耦合的非均匀热方程，提出Dirichlet--Neumann波形松弛的全离散$l^2$误差分析，揭示CFL数对收敛性的影响，并通过数值实验验证估计的准确性。
AI 中文摘要
我们考虑两个在不同空间域上通过低维界面相互作用的耦合线性热方程。这模拟了共轭热传递。该问题使用Dirichlet--Neumann波形松弛方法求解，允许子问题使用独立的代码求解，即所谓的分区方法。我们的总体目标是开发更高效的分区方法，为此，我们需要可靠的误差估计。在这里，我们使用指数加权傅里叶技术，在全离散设置下，针对有限时间$T$，推导出$l^2$中新的误差估计。这些估计描述了线性和超线性行为。我们表明，当子求解器中的CFL数较大时，全离散估计接近先前获得的时间离散估计，并且与$\u0394x_1$和$\u0394x_2$无关。我们还表明，当CFL数较小时，收敛行为取决于比值$\u0394x_1/\u0394x_2$。我们的数值实验表明，全离散估计在广泛的网格尺寸$\u0394x_1, \u0394x_2$、时间步长$\u0394t$和$T$范围内是准确的。
英文摘要
We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet--Neumann waveform relaxation. This allows the subproblems to be solved using separate codes, a so called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. Here, we use an exponentially weighted Fourier technique to derive new error estimates in $l^2$ for finite time $T$ in the fully discrete setting. These describe both linear and superlinear behavior. We show that the fully discrete estimate is close to a previously obtained time discrete estimate and independent of $Δx_1$ and $Δx_2$ when the CFL number in the subsolvers is large. We also show that the convergence behaviour depends on the ratio $Δx_1/Δx_2$ when the CFL number is small. Our numerical experiments show that the fully discrete estimate is accurate across a wide range of grid sizes $Δx_1, Δx_2$, time step sizes $Δt$ and $T$.
