摘 要：数值积分方法在物理问题求解中具有重要地位，其精度直接影响计算结果的可靠性。本研究以提高数值积分精度为目标，针对不同物理场景下的积分问题，系统评估了多种经典数值积分方法的适用性与精度特性。通过引入自适应步长控制和高阶积分公式，提出了一种改进型复合积分算法，有效降低了传统方法在处理非线性或奇异函数时的误差。研究采用理论分析与数值实验相结合的方式，对典型物理模型进行了验证，包括波动方程、热传导方程及量子力学中的薛定谔方程等。结果表明，改进算法在保持计算效率的同时显著提升了积分精度，尤其在处理复杂边界条件和高频振荡函数时表现出优越性能。本研究的主要贡献在于为物理问题的数值求解提供了更为精确和高效的积分工具，同时为相关领域算法优化提供了理论支持与实践参考。
关键词：数值积分；精度优化；自适应步长；复合积分算法；物理模型验证
Accuracy Assessment of Numerical Integration Methods in Solving Physical Problems
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Abstract：Numerical integration methods play a crucial role in solving physical problems, and their accuracy directly affects the reliability of computational results. This study aims to enhance the precision of numerical integration by systematically evaluating the applicability and accuracy characteristics of various classical numerical integration methods for integration problems under different physical scenarios. By incorporating adaptive step size control and higher-order integration formulas, an improved composite integration algorithm is proposed, which effectively reduces errors encountered in traditional methods when dealing with nonlinear or singular functions. The research combines theoretical analysis with numerical experiments to validate typical physical models, including the wave equation, heat conduction equation, and Schrödinger equation in quantum mechanics. The results demonstrate that the improved algorithm significantly enhances integration accuracy while maintaining computational efficiency, particularly excelling in handling complex boundary conditions and high-frequency oscillatory functions. The primary contribution of this study lies in providing a more precise and efficient integration tool for the numerical solution of physical problems, as well as offering theoretical support and practical references for algorithm optimization in related fields.
Keywords: Numerical Integration；Precision Optimization；Adaptive Step Size；Composite Integration Algorithm；Physical Model Verification
目 录
引言 1
一、数值积分方法概述 1
(一)常用数值积分方法简介 1
(二)数值积分的基本原理分析 2
(三)物理问题中的数值积分需求 2
二、精度评估的理论基础 3
(一)误差来源与分类研究 3
(二)精度评估的核心指标体系 3
(三)影响精度的关键因素分析 4
三、典型物理问题的数值求解 4
(一)力学问题中的数值积分应用 4
(二)电磁场计算的精度优化策略 5
(三)热传导问题的数值模拟分析 5
四、精度改进与优化方法 6
(一)高阶数值积分方法探讨 6
(二)自适应算法在精度提升中的作用 6
(三)复合积分法的应用与效果评估 7
结论 7
参考文献 8
致谢 8
