摘 要：热传导问题是工程与科学领域中广泛存在的基本问题，其数学描述通常归结为偏微分方程的求解。本文旨在探讨基于有限差分法和有限元法的数值解法在热传导问题中的应用，并提出一种改进的高精度离散格式以提高计算效率和精度。研究通过构建非均匀网格下的修正差分算子，有效解决了传统方法在处理复杂边界条件时的误差累积问题，同时结合自适应时间步长策略优化了动态传热过程的模拟效果。数值实验结果表明，所提方法在保持较高精度的同时显著降低了计算成本，尤其在非稳态热传导问题中表现出优越的收敛特性。此外，本文还验证了该方法在多材料界面传热问题中的适用性，为实际工程应用提供了可靠的理论支持。本研究的主要贡献在于提出了一种兼顾效率与精度的数值算法，为复杂热传导问题的高效求解提供了新思路。
关键词：热传导问题；有限差分法；有限元法；高精度离散格式；非均匀网格
Numerical Solution of Partial Differential Equations in Heat Conduction Problems
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Abstract：Heat conduction is a fundamental problem widely encountered in engineering and scientific fields, and its mathematical desc ription typically involves the solution of partial differential equations. This paper investigates the application of numerical methods based on the finite difference method and the finite element method in solving heat conduction problems, proposing an improved high-precision discretization scheme to enhance computational efficiency and accuracy. By constructing a modified difference operator under non-uniform grids, the study effectively addresses the issue of error accumulation in traditional methods when dealing with complex boundary conditions, while combining an adaptive time-stepping strategy to optimize the simulation of dynamic heat transfer processes. The results of numerical experiments demonstrate that the proposed method significantly reduces computational costs while maintaining high accuracy, particularly exhibiting superior convergence characteristics in non-steady-state heat conduction problems. Furthermore, the applicability of this method in multi-material interface heat transfer problems is verified, providing reliable theoretical support for practical engineering applications. The primary contribution of this research lies in the development of a numerical algorithm that balances efficiency and precision, offering a new approach for the efficient solution of complex heat conduction problems.
Keywords: Heat Conduction Problem；Finite Difference Method；Finite Element Method；High Accuracy Discrete Scheme；Non-Uniform Grid
目 录
引言 1
一、热传导问题的数学建模 1
(一)偏微分方程的基本形式 1
(二)热传导方程的推导过程 2
(三)边界条件与初始条件设定 2
二、数值解法的基础理论 3
(一)有限差分法的基本原理 3
(二)稳定性与收敛性分析 3
(三)离散格式的选择与优化 4
三、数值方法在热传导中的应用 4
(一)显式格式的实现与特点 4
(二)隐式格式的设计与求解 5
(三)混合格式的应用场景 5
四、实际问题中的数值模拟 6
(一)复杂边界条件的处理 6
(二)非稳态热传导的计算 6
(三)结果验证与误差分析 7
结论 7
参考文献 9
致谢 9
