In this paper, a quadratic time interpolation for temperature and a linear time interpolation for fluxes are implemented for the parabolic (time-dependent) fundamental solution-based scheme for solving transient heat transfer problems with sources using the subdomain BEM (boundary element method), which is the main innovation of this paper. The approach described in this work to incorporate the quadratic time variation does not require doubling the number of equations, which is otherwise required in the BEM literature, for the discretized problem to be well-conditioned. Moreover, the numerical accuracy, compared over an unprecedented range of the Fourier number (Fo) and source strength values, can help in selecting the appropriate scheme for a given application, depending on the rate of the heat transfer process and the included source term. The newly implemented scheme based on the parabolic fundamental solution is compared with the well-established elliptic (Laplace) scheme, where the time derivative of the temperature is approximated with the second-order finite difference scheme, on two examples.

中文翻译：

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用于传热建模的边界元中瞬态基本解的二次时间单元


本文对基于抛物线（瞬态）基本解的方案实施了温度二次时间插值和磁通量线性时间插值，该方案使用子域 BEM（边界元法）求解源瞬态传热问题，这是本文的主要创新。这项工作中描述的合并二次时间变化的方法不需要将方程的数量增加一倍，而 BEM 文献中则要求将离散化问题很好地调节。此外，在傅里叶数 （Fo） 和源强度值的历史新范围内进行比较的数值精度有助于为给定应用选择合适的方案，具体取决于传热过程的速率和包含的源项。在两个示例中，将基于抛物线基本解的新实现方案与成熟的椭圆（拉普拉斯）方案进行了比较，其中温度的时间导数用二阶有限差分方案进行近似。