Course Overview

This is the first course in numerical analysis for graduate students. The main objectives of this course include: (1) development and applications of numerical methods when analytical techniques are not available; (2) development of a conceptual framework for analysis of methods to fix the problem; (3) discrete calculus and approximations; (4) tradeoffs between accuracy and computational cost;

Learning Achievement

1. Interpolation (3 hrs)
(1) Lagrange Polynomials
(2) Polynomial Interpolations; Splines
2. Numerical Differentiation (4 hrs)
(1) Construction of Finite Difference Scheme, Order of Accuracy
(2) Modified Wavenumber as a Measure of Accuracy
(3) Pade Approximation
(4) Matrix Representation of Finite Difference Schemes
3. Numerical Integration (8 hrs)
(1) Trapezoidal Rule; Simpson’S Rule; Error Analysis and Mid-Point Rule
(2) Romberg Integration and Richardson’S Extrapolation
(3) Adaptive Quadrature; Gauss Quadrature
4. Numerical Solution of Ordinary Differential Equations (10 hrs)
(1) Initial Value Problems; Numerical Stability Analysis, Model Equation
(2) Accuracy; Phase and Amplitude Errors
(3) Runge-Kutta Type Formulas, Multi-Step Methods; Implicit Methods
(4) System of Differential Equations; Stiffness
(5) Linearization For Implicit Solution of Non-Linear Differential Equations
(6) Boundary Value Problems, Shooting, Direct Methods, Non-Uniform Grids, Eigenvalue Problems
5. Partial Differential Equations (10 hrs)
(1) Finite-Difference Solution of Partial Differential Equations
(2) Modified Wavenumber and Von Neumann Stability Analysis, Modified Equations Analysis
(3) Alternating Direction Implicit Methods; Non-Linear Equations; Iterative Methods for Elliptic Pde's

Competence

Course prerequisites

HOMEWORKS (55%); MIDTERM EXAM (%15); FINAL EXAM (%30)

Grading Philosophy

Course schedule

Course type

Online Course Requirement

Instructor

Chou, Yi-Ju

Other information

(College of Engineering) Graduate Institute of Applied Mechanics

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