Courses Detail Information
MATH6003J – Introduction to Engineering Numerical Analysis
Instructors:
Credits: 3 credits
Pre-requisites: Graduate Standing
Description:
The course gives an introduction to the wide range of basic and advanced numerical approaches that might be used to solve engineering problems related to differential equations. The explanation of relevance, limitations and possible modifications of numerical methods is the primary focus of the course. Some convergence theorems and error estimates are discussed and proved. In the last part of the course we will work with recent results in the field of numerical approximations, students will read and discuss various articles and possibly extend some of the existing results if interested. The course also provides practice in computer programming using MATLAB/Maple.
Course Topics:
Introduction. Taylor series and finite difference approximations.
The first order Euler methods. Consistency, stability, convergence.
Linearization of nonlinear implicit FDEs.
Single point methods. Error estimation and error control.
Extrapolation and multipoint methods. Systems of first-order ODEs. Stiff ODEs.
The shooting (initial-value) method.
The equilibrium (boundary-value) method.
Higher-order methods and non-uniform grids.
Introduction to PDEs and classification of physical problems.
Approximation by finite differences, truncation error. The finite difference method for the Laplace (Poisson) equation.
Consistency, stability, convergence.
Elliptic PDEs: iterative methods, derivative boundary conditions.
Higher-order methods for elliptic PDEs.
Nonrectangular domains.
Parabolic PDEs: finite difference methods, consistency, stability, convergence.
Parabolic PDEs: the Richardson and Dufort-Frankel methods. Implicit methods.
Parabolic PDEs: nonlinear and multidimensional problems.
Hyperbolic PDEs: FTCS method and the Lax method
Introduction to finite element methods (FEMs): the Rayleigh-Ritz, collocation and Galerkin methods.
FEM for boundary-value problems. FEM for the Laplace equation. FEM for diffusion problems.
Review of advanced approaches in numerical solutions of differential equations.
Review of advanced approaches in numerical solutions of differential equations.