Analysis of numerical methods for differential equations

• Grant number: 17/14131-0
• Support Opportunities: Regular Research Grants
• Start date: September 01, 2017
• End date: August 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Messias Meneguette Junior
• Grantee: Messias Meneguette Junior
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Associated researchers: Almir Olivette Artero ; Analice Costacurta Brandi ; Vanessa Avansini Botta Pirani

Abstract

The area of numerical methods for differential equations is wide and has received increasing attention due to the possibility of numerical simulations of problems and geometries of greater complexity. A key aspect is the analysis of these methods. Under the broader framework analysis of numerical methods for differential equations we are investigating: 1) numerical stability of the family of generalized BDF methods, made in theoretical basis involving results on the roots of polynomials; 2) analysis of the numerical difficulties of the HWNP (high Weissenberg number problem) whose results have shown the need for methods with properties that provide guarantees of adequate growth of the numerical solution and our initial investigation indicates that the well-balanced methods are promising; as they were not yet introduced for the flow simulations of viscous-elastic materials we shall use them and compare their performance against established methods, such as the Log Conformation Representation (LCR); 3) analysis of theoretic-numerical aspects of SPH (and variants) and MPS methods (meshless), for which many details have to be improved, for example, boundary treatment, consistency order, suitability for more complex flow simulation, implementation on GPU, etc. A major motivation is the simulation of more complex flows, for example, including non-Newtonian models, free boundary, etc.; so we are creating conditions for progressing in this direction. (AU)