The quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the “mathematical microscope”, has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low-pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet-based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics.

中文翻译：

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小波方法在计算物理中的应用

许多物理问题的定量研究最终都归结为求解各种偏微分方程（PDE）。小波分析被誉为“数学显微镜”，因其出色的多分辨率分析（MRA）能力及其基函数而被誉为“数学显微镜”，其基函数具有正交性、紧支撑、低通滤波和插值等各种理想的数学特性。这些特性使小波成为有效求解这些偏微分方程的强大工具。30多年来，小波伽辽金法、小波配置法、小波有限元法、小波积分配置法等数值方法被提出并成功应用于各种物理问题的定量分析。本文将从小波MRA的基本理论出发，简要总结各种基于小波基的数值方法的优点和局限性。本文的目的是帮助研究人员针对其特定的物理问题选择合适的数值方法。此外，它将探索基于小波的技术的前瞻性进展，为致力于增强计算物理领域小波数值方法的研究人员提供宝贵的见解。