Review of Collocation Methods and Applications in Solving Science and Engineering Problems

Abstract

The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations. This paper provides a comprehensive review of collocation methods and their applications, focused on elasticity, heat conduction, electromagnetic field analysis, and fluid dynamics. The merits of the collocation method can be attributed to the need for element mesh, simple implementation, high computational efficiency, and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method. Beginning with the fundamental principles of the collocation method, the discretization process in the continuous domain is elucidated, and how the collocation method approximation solutions for solving differential equations are explained. Delving into the historical development of the collocation methods, their earliest applications and key milestones are traced, thereby demonstrating their evolution within the realm of numerical computation. The mathematical foundations of collocation methods, encompassing the selection of interpolation functions, definition of weighting functions, and derivation of integration rules, are examined in detail, emphasizing their significance in comprehending the method’s effectiveness and stability. At last, the practical application of the collocation methods in engineering contexts is emphasized, including heat conduction simulations, electromagnetic coupled field analysis, and fluid dynamics simulations. These specific case studies can underscore collocation method’s broad applicability and effectiveness in addressing complex engineering challenges. In conclusion, this paper puts forward the future development trend of the collocation method through rigorous analysis and discussion, thereby facilitating further advancements in research and practical applications within these fields.