Abstract In this paper, the backward heat conduction problem is solved numerically using a fourth-order compact difference scheme. For the regularization of this ill-posed problem, fourth-order compact filtering which is a spatial filtering technique is proposed. The second derivative approximation in compact difference scheme has been compared with the fourth-order standard finite difference scheme via Fourier analysis. The comparison shows that the resolution ability of the compact difference scheme is better than the standard finite difference scheme. Meanwhile, an error estimate between the exact and regularized solution is derived. Numerical results are provided using the proposed method. The comparison of CPU time shows that the fourth-order compact difference scheme takes lesser CPU time than the fourth-order standard finite difference scheme.

中文翻译：

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紧凑滤波作为反向热传导问题的正则化技术

摘要 本文采用四阶紧致差分格式数值求解后向热传导问题。为了正则化这个不适定问题，提出了作为空间滤波技术的四阶紧凑滤波。通过傅里叶分析，将紧凑差分格式中的二阶导数近似与四阶标准有限差分格式进行了比较。对比表明，紧致差分方案的分辨能力优于标准有限差分方案。同时，推导出精确解与正则化解之间的误差估计。使用所提出的方法提供数值结果。