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Dr. Noemi Friedman, 28.10.2015.
Introduction to PDEs and Numerical Methods
Lecture 1:
Introduction
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Basic information on the course
Course Title:
Introduction to PDEs and Numerical Methods
Lecturer:
Nomi Friedman
Mhlenpfordtstr. 23, 8th floor
Room: 819
Assistant (exercises):
Jaroslav Vondejc
Mhlenpfordtstr. 23, 8th floor
Room: 822
Assistant2 (small tutorials):
Stephan Lenz (CSE student)
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Basic information on the course
Credits and work load:
5 credits: 6-7 hours/week
Pre-requisits:
Differential operators,
elementary knowledge of PDEs,
basics of linear algebra,
basic MATLAB coding skills
Requisits:
Weekly assignments in group of two or three (min 50%)
Written exam: 29.2.2016., 10:30 - 12:00, room ZI 24.1
Script, recommended literature:
See webpage: https://www.tu-braunschweig.de/wire/lehre/ws15/pde1
Software used:
MATLAB, FEniCS (Python interface)
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Information about the assignments
Homework assignments in groups (max. group of three)
Submission of homework
Written homework
Submit on the beginning of the tutorial
(include cover sheet with subject name (PDE1), names and matriculation
number of students, assignment number)
For program codes:
e-mail: [email protected]
subject: assignment# NAMES
(#: number of the assignment, NAMES: names of students)
(e.g.: assignment1 J. Smith, K. Park)
Homework is due to the beginning of the tutorials
Consultation:
Noemi Friedman (after the lecture, office hours by arrangement, please, take
appointment first by e-mail: [email protected])
Jaroslav Vondejc (will be assigned on the tutorial
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Definition: ODEs PDEs
Partial differential equation:
Equation specifying a relation between the partial derivative(s) of an unkown
multivariable function and maybe the function itself:
, , , , , , ,
, , , ,
,
2 , , ,
, = (, , , )
Ordinary differential equation:
Equation specifying a relation between the derivative(s) of an unkown univariable
function and maybe the function itself:
,
,
2
2, = ()
Boundary Value Problem (BVP), Initial Boundary Value Problem (IBVP):
PDE with initial/boundary conditions
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Motivation simulation of planets
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Motivation heat convection
Source:
ANSYS http://www.ansys.com/staticassets/ANSYS/staticassets/product/16-highlights/underhood-simulation-surface-temps-heat-
transfer-manifold-2.jpg
computed surface temperatures due to
convective and radiative heat transfer from
the exhaust manifold to surrounding
objects
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Motivation structural analysis
Source: ANSYS
http://wildeanalysis.co.uk/system/photos/838/preview/ansys_ex
plicit_str.png?1273430962
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Motivation flow problems
The Stokes equation
Source:
FENICS documentation:
http://fenicsproject.org/documentation/dolfin/1.6.0/python/demo/documented/stokes-taylor-hood/python/documentation.html
+ = in = 0 in
(x)(x)
Motivation flow problems
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Source: https://33463d8ba37cd0a930f1-
eb07ed6f28ab61e35047cec42359baf1.ssl.cf5.rackcdn.com/ugc/entry/5
44867a78ef07-133981577813387476428_0129_fast15.jpg
Source: TU Braunschweig SFB 880
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Motivation highly coupled systems
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Overview of the course
Introduction (definition of PDEs, classification, basic math,
introductory examples of PDEs)
Analitical solution of elementary PDEs (Fourier series/transform,
seperation of variables, Greens function)
Numerical solutions of PDEs:
Finite difference method
Finite element method
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Overview of this lecture
Basic definitions, motivation
Differential operators: basic notations, divergence, Laplace, curl, grad
Classification of PDEs
Introductory example: heat flow in a bar
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Differential operators partial derivative
Partial derivative:
(, , , )
, , ,
, , , (
)
(Image source: Wikipedia)
= (, ) = 2 + + 2
= 2 +
=1,=1
= 3
=
Example:
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Differential operators mixed derivative
Mixed derivative:
(, )
2
=
2
(, , ) = 2cos()
= 2cos()
= 2 cos()
2
= 2cos()
2
= 2cos()
Example:
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Differential operators total derivative
Total derivative:
Total differential (differential change of f):
=
+
+
+
= , , ,
=
+
+
+
, = 2 + 2
=
+
= 2 + 2 =
= 2
total derivative
partial derivative
Example:
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Differential operators gradient
Nabla operator:
Gradient:
= , , : 3 (vector-scalar function)
=
Example:
direction: greatest rate of increase of
the function
magnitude: the slope of the function in
that direction
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Differential operators directional derivative
= , , : 3 (vector-scalar function)
= lim0
+
= = =
, ,
, ,
, ,
Directional derivative:
(normalised):
= lim0
+
=
What is the differential equation to define a wave
traveling with speed ?In the direction is constant directionalderivative is zero:
, =
, = 0 ( , = 0)
Example 1:
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Differential operators directional derivative
, = , = 1
= 0
=1 , = 0
Lets suppose = sin( ) is a solution of the transport equation.What is its directional derivative in the direction:
+ = 0Transport equation:
=1
, = , = 1cos( )
cos( = 0
Example 2:
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Differential operators - divergence
of (, , ): 3 3((of a vector field):
, , =
(, , )(, , )
(, , )
=
, , = , , =
=
+
+
Example:
Divergence:
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Differential operators - Laplace
Example:
Laplace operator:
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Differential operators rotation (curl)
of (, , ): 3 3((of a vector field):
, , =
(, , )(, , )
(, , )
=
, , = , , =
=
Example:
Rotation (curl):
direction: axis of rotation
magnitude: magnitude of rotation
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Classification of PDEs
Constant/variable coefficients
Stationary/instationary (not time dependent/time dependent)
Linear/nonlinear
linearity condition:
order
order of the highest derivative
homogeneous/inhomogenous
inhomogeneous: additive terms which do not depend on unknown function
homogenous: = 0 is a solution of the equation elliptic/parabolic/hyperbolic (only for second order PDEs)
+ 2 + + lower order derivatives = 0
2 = 0 parabolic 2 < 0 hyperbolic 2 > 0 elliptic
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Classification of PDEs, examples of PDEs
Wave equation
= 0
Laplace equation
+ = 0Heat equation
= 0
Order 2 2 2
Constant
coefficient?
yes yes yes
Linear? yes yes yes
Homogenous? yes yes yes
Class A=1, B=0, C=
2 =
Hyperbolic
A=1, B=0, C=1 2 =1
Elliptic
A=-1, B=0, C=0 2 = 0
Parabolic
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Classification of PDEs, examples of PDEs
Linearity:
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Classification of PDEs, examples of PDEs
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Introductory example: heat flow in a bar
Basic assumptions:
Uniform cross-section
Temperature varies only in the longitudinal direction
Relationship between heat energy and temperature is linear
: cpecific heat capacity Energy []
energy is required to raise the tempreture by 1 of 1 materialHomogenous material properties along the bar ( and are constants along the bar)
[
3]: density of the material of the bar
Problem description , =? (temparature)
0 +
(0, ) (, )
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Introductory example: heat flow in a bar
0 +
(0, ) (, )
Total energy in the bar section of length is:
= 0 +
+
( , 0)) =
0 +
+
,
+
0
0But Im only interested in:
+
, =
+
,
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Introductory example: heat flow in a bar
+
, =
+
,
Change of heat energy by time in the section of length
(, ) ( + , )
Change of energy by time in the section of length from heat flux:
,
2: heat flux
+ , , =
+
,
Fundamental
theorem of calculus:
Conservation of energy:
+
,
=
+
,
+
,
+
,
= 0
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Introductory example: heat flow in a bar
(, ) ( + , )
+
,
+
,
= 0
,
+
,
= 0 0 < <
,
: change of temperature with increasing
Assumption: Fouriers law of heat conduction: , = ,
,
2 ,
2= 0Heat equation
()() ,
()
,
= 0 (for inhomogenous material
properties)
(for homogenous material properties)
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Introductory example: heat flow in a bar
(, ) ( + , )
+
,
+
,
=
+
(, )
,
+
,
= (, ) 0 < <
,
2 ,
2= (, )
Heat equation
()() ,
()
,
= (, ) (for inhomogenous material properties)
(for homogenous material properties)
The heat equation with source or sink (inhomogenous heat equation)
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Introductory example: heat flow in a bar
0
(0, ) (, )
,
2 ,
2= (, )
Boundary conditions:
a) Perfect isolation at the end (flux across the boundaries is zero):
0,
= 0
,
= 0
b) Perfect thermal contact:
0, = 0 , = 0