Some practical remarks in solving partial differential equations using reduced order schemes obtained through the POD method

• Published in: INCAS bulletin Vol. 14; no. 1; pp. 187 - 196
• Main Authors: SOLOMON, Alexandru, OLTEI, Valentin Claudiu, BOGOI, Alina
• Format: Journal Article
• Language: English
• Published: Bucharest INCAS - National Institute for Aerospace Research "Elie Carafoli" 07.03.2022; National Institute for Aerospace Research “Elie Carafoli” - INCAS
• Subjects: Algorithms; Decomposition; Eigenvalues; Exact solutions; Finite difference method; Model reduction; Optimization; Partial differential equations; Proper Orthogonal Decomposition; proper orthogonal decomposition (pod); reduced order method (rom); singular value decomposition (svd); Truncation errors
• ISSN: 2066-8201; 2247-4528
• DOI: 10.13111/2066-8201.2022.14.1.15

In this paper we address the subject of mathematical modelling, more precisely the optimization of algorithms for numerically solving partial differential equations. The problem proposed to be tackled in this paper is the implementation of an algorithm for solving partial differential equations in a significantly faster way than that obtained through applying finite difference schemes. The proper orthogonal decomposition (POD) method is a modern and efficient method of reducing the number of variables that occur as a result of applying centred difference schemes to partial differential equations, thus reducing the running time of the algorithm and the accumulation of truncation errors. Therefore, the POD method has been implemented to obtain a reduced order scheme applied to different partial differential equations, with some practical applications and comparisons with the analytical solutions.