Studies and implementations of numerical methods for differential equations using finite difference technique and applications in computational fluid mechanics

• Grant number: 11/20786-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: January 01, 2012
• End date: December 31, 2012
• Field of knowledge: Engineering - Mechanical Engineering - Transport Phenomena
• Principal Investigator: Gilcilene Sanchez de Paulo
• Grantee: Caroline Viezel
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil

Abstract

In this project, we intend to study and implement numerical methods for ordinary differential equations (ODE) and partial differential equations (PDE) using the technique of finite differences. Initially, the student will study some numerical methods for ODE and the concept of stability. In addition, we intend to study the different effects that each numerical boundary condition causes the problem. As motivation to study computational fluid mechanics intends to apply a numerical technique for the problem of the simplified boundary layer (toy model). Later, the student will study some numerical methods for PDE and the concepts of stability and convergence for explicit, implicit, and Crank-Nicolson methods through the heat equation. As an application, the student should simulate the problem of diffusion of heat through a material such as solid metal. Also, it is intended to consider a nonlinear equation of convection-diffusion to study the effects of convection on the numerical solution and some classical numerical methods that can treat such terms. Finally, it is expected that the student get a short knowledge of rheological equations and their importance in the numerical simulation of viscoelastic fluid flows.(AU)