2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 5
REVIEW Lecture 4
Roots of nonlinear equations: Open Methods
Fixed-point Iteration (General method or Picard Iteration), with examples
Iteration rule:
xn 1 g ( xn )
xn 1 x h( xn ) f ( xn )
n
or
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 18
REVIEW Lecture 17:
Stability (Heuristic, Energy and von Neumann)
Hyperbolic PDEs and Stability, CFL condition, Examples
Elliptic PDEs
FD schemes: direct and iterative
Iterative schemes, 2D: Laplac
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 19
REVIEW Lecture 18:
Finite Volume Methods
Integral and conservative forms of the cons. laws
Introduction
Approximations needed and basic elements of a FV scheme
Time-Marching and Grid generation
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 9
REVIEW Lecture 8:
Direct Methods for solving linear algebraic equations
Gauss Elimination
Algorithm
Forward Elimination/Reduction to Upper Triangular System
Back-Substitution
Number of Operation
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 16
REVIEW Lecture 15:
Fourier Error Analysis
Provide additional information to truncation error: indicates how well Fourier mode
solution, i.e. wavenumber and phase speed, is represented
Effective wa
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 20
REVIEW Lecture 19: Finite Volume Methods
Review: Basic elements of a FV scheme and steps to step-up a FV scheme
One Dimensional examples
d x
f
Generic equation:
j
dt
j 1/ 2
f j 1/ 2
x j 1/ 2
x
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 22
REVIEW Lecture 21:
End of Time-Marching Methods: higher-order methods
Runge-Kutta Methods
Additional points are between tn and tn+1
Multistep/Multipoint Methods: Adams Methods
n 1
n
tn 1
f (t, ) dt
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 25
REVIEW Lecture 24:
Solution of the Navier-Stokes Equations
v
.( v v ) p 2v g
t
.v 0
Discretization of the convective and viscous terms
Discretization of the pressure term
2
2 u
p p g.r .u ( p ei gi
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 14
REVIEW Lecture 12-13:
Classification of PDEs and examples of finite-difference discretization
Error Types and Discretization Properties
Finite Differences based on Taylor Series Expansions
High
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 15
REVIEW Lecture 14:
Finite Difference: Boundary conditions
Different approx. at and near the boundary => impacts linear system to be solved
Finite-Differences on Non-Uniform Grids and Uniform Err
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 6
REVIEW of Lecture 5
Continuum Hypothesis and conservation laws
Macroscopic Properties
Material covered in class: Differential forms of conservation laws
Material Derivative (substantial/total derivativ
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 4
REVIEW Lecture 3
Truncation Errors, Taylor Series and Error Analysis
x 2
x 3
x n n
Taylor series: f ( xi 1 ) f ( xi ) x f '( xi )
f '( xi )
f '( xi ) .
f ( xi ) Rn
2!
Rn
n!
3!
n 1
x
f ( n 1)
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 3
REVIEW Lectures 1-2
Approximation and round-off errors
Absolute and relative errors: Ea xa x , a
Iterative schemes and stop criterion:
For n digits:
a
1
2
s 10 n
xa x
xa
xn xn 1
s
xn
Number re
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 10
REVIEW Lecture 9:
Direct Methods for solving linear algebraic equations
Gauss Elimination
LU decomposition/factorization
Error Analysis for Linear Systems and Condition Numbers
Special Matri
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 11
REVIEW Lecture 10:
Direct Methods for solving (linear) algebraic equations
Gauss Elimination
LU decomposition/factorization
Error Analysis for Linear Systems and Condition Numbers
Special Matric
2.29 Numerical Fluid Mechanics Fall 2011 Lecture 7
REVIEW of Lecture 6
Material covered in class: Differential forms of conservation laws
Material Derivative (substantial/total derivative)
Conservation of Mass
Differential Approach
Integral (volume) A
2.29 Numerical Fluid Mechanics
Fall 2011 Lecture 8
REVIEW Lecture 7:
Direct Methods for solving Linear Equation Systems
Determinants and Cramers Rule (and other methods for a small
number of equations)
Gauss Elimination
Algorithm
Forward Eliminat
Department of Applied Mathematics
Computer Assignment 1
Instructions: Use SPSS or other statistical packages to analyze the data of the assigned
problem.
Answer the questions based on the computer outputs and draw a brief
conclusion. Hand calculation, lis