Numerical Analysis (I)

Projects

1. (Due 10/30/98) Newton's Method (I): Demo
2. (Due 11/13/98) Matrix Computations (I): Demo
• Read the example (7.2.12) in Patel p.180 and solve it by adaptc++
• Solve C.8  in Patel p.186 by adaptc++
3. CParabProb (I): 1D Convection-Diffusion-Reaction (Parabolic) Problems
• (Due 11/20/98) Complete the program xmain-heat1.cxx (Do NOT compile and run this program. Just fill up the blank member functions.)
• (Due 12/4/98) Given xparabprob.hpp. Complete the programs   xmain-heat1.cxx, xmain-heat2.cxx, and xparabprob.cxx (Add these files to adapt98, then build, and execute it.)
4. CParabProb (II): 1D Convection-Diffusion-Reaction (Parabolic) Problems
• (Due 1/8/99) Revise the programs   xmain-heat1.cxx, xmain-heat2.cxx, and xparabprob.cxx
5. Lecture note writing: (Due 1/15/99)
• Download sample files: tranport.tex, imeshf.gif, imesh.gif, fmesh.gif.

Exams

1. Midterm Exam  on 11/23/98
2. Final Exam 1/18/99

Lectures

1. Computer Arithmetic (Chap 1) 10/12
2. Newton's Method (Chap 2) 10/19
• Convergence Properties
• Taylor's Theorem
3. Scientific Computing -- An Over View
• Nature -- Biology, Chemistry, Physics*, and Sociology
• Law of Mass Conservation
• Law of Energy Conservation
• Law of Momentum Conservation
• Mathematical Modeling
• Algebraic Equations
• Ordinary Differential Equations (ODEs)
• Partial Differential Equations (PDEs)*
• Time Dependent Problems*
• Heat Equation* (Parabolic Type)
• Wave Equation (Hyperbolic Type)
• Time Independent Problems
• Laplace's Equation* (Elliptic Type)
• Integral Equations
• Mixed Equations
• Differential Algebraic Equations
• Integral Differential Equations
• Numerical Computations
• Numerical Differentiation
• Numerical Integration
• Numerical Linear Algebra* (Ax=b)
• Numerical Methods for Nonlinear Equations*
• Newton's Method
• Monotone Method
• Numerical Methods for ODEs
• Numerical Methods for PDEs*
• Time Discretization
• Forward Difference Method -- Explicit Method
• Central Difference Method -- Richardson's Method
• Backward Difference Method -- Implicit Method
• The Crank-Nicolson Method
• Spatial Discretization
• Finite Difference Method (FDM)
• Finite Element Method (FEM)
• Finite Volume Method (FVM)
4. A Mathematical Model
• Convection-Diffusion-Reaction (CDR) Model
• Convection: Together + Carry -- Transfer by movement.
• Diffusion: Apart + Flow -- Spreading out in every direction.
• Reaction: Response to a stimulus or influence.
• Applications
• Heat Conduction
• Highway Traffic
• River Pollution
• Semiconductor
• Modeling: Conservation of Mass & Mean Value Theorem
• Space variable: x (unit L), time variable: t (unit T)
• Substance: Heat, cars, pollutants, algae etc. (unit M)
• u=u(x,t): The density or concentration of a substance at x and at time t in units of mass per unit distance (M/L).
• q=q(x,y): Its rate of flow past x at time t in units of mass per unit time (M/T).
• f=f(x,t): The rate at which the desity of mass is changing due to source, sink, or reaction (M/(LT)).
• v=v(x,t): The velocity of the flow in whch the substance moves (L/T).
• Convection: q=vu
• Diffusion: q=- c (du/dx), c>0, a constant of proportionality.
• Reaction: f.
• Equation: du/dt = c {d^2u/dx^2} - {d(vu)/dx} + f
5. The Finite Didderence Method for the Heat Equation
• Problem
• Heat Equation
• Boundary Conditions (BCs)
• Initial Condition (IC)
• Numerical Methods
• Time Discretization by FDM
• The Forward Difference Method
• The Backward Difference Method
• The Central Difference (Crank-Nicalson) Method
• Space Discretization by FDM
• The Central Difference Method => Stiffness Matrix S
• Patel p.139 Table 5.3.1
• Full Discretization in Matrix Form
• Impose BCs and IC => Ax=b
6. The Finite Didderence Method for Convection-Diffusion-Reaction Problems
• Space Discretization by FDM
• Stiffness Matrix S
• Load Vector F
• Full Discretization in Matrix Form
• Impose BCs and IC => Ax=b
7. The Finite Element Method for the Heat Equation
• Weak Formulation
• Integration by Parts
• Test and Trial Functions
• Function Spaces
• Finite Element Spaces
• Semi-Discrete Finite Element Approximation

Text

Vithal A. Patel, Numerical Analysis, Saunders College Publishing, 1994.

Syllabus

Chapters 1, 2, 7, 11, 12, 13, 14, 15