2022/07/12 by Jonna C. Roden, Rory D. Mills-Williams, Roden, Jonna C. +5
Computer Science · Mathematics · #35Q70 #35Q93 #65M70 #82C22 #82M22 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · doi:10.48550/arxiv.2207.05589
openalex publication_date 2022/07/12 · openalex created_date 2022/07/15 · openalex updated_date 2026/07/28
A numerical framework is developed to solve various types of PDEs on complicated domains, including steady and time-dependent, non-linear and non-local PDEs, with different boundary conditions that can also include non-linear and non-local terms. This numerical framework, called MultiShape, is a class in Matlab, and the software is open source. We demonstrate that MultiShape is compatible with other numerical methods, such as differential--algebraic equation solvers and optimization algorithms. The numerical implementation is designed to be user-friendly, with most of the set-up and computations done automatically by MultiShape and with intuitive operator definition, notation, and user-interface. Validation tests are presented, before we introduce three examples motivated by applications in Dynamic Density Functional Theory and PDE-constrained optimization, illustrating the versatility of the method.