tobias bleninger matemÁtica aplicada i
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1
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
MATEMÁTICA APLICADA ITobias Bleninger
2
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Source:
Chapra, Numerical
Methods for Engineers,
2008
3
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Differentiation
The mathematical definition of a derivative begins with a difference
approximation:
and as x is allowed to approach zero, the difference becomes a
derivative:
y
x=f xi + x( )− f xi( )
x
dy
dx= lim
x→0
f xi + x( )− f xi( )x
Source:
Chapra, Numerical Methods for Engineers, 2008
4
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
High-Accuracy Differentiation Formulas
Taylor series expansion can be used to generate high-accuracy formulas for
derivatives by using linear algebra to combine the expansion around several
points.
Three categories for the formula include forward finite-difference, backward
finite-difference, and centered finite-difference.
Source:
Chapra, Numerical Methods for Engineers, 2008
5
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Forward Finite-Difference
Source: Chapra, Numerical Methods for Engineers, 2008
6
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Backward Finite-Difference
Source: Chapra, Numerical Methods for Engineers, 2008
7
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Centered Finite-Difference
Source: Chapra, Numerical Methods for Engineers, 2008
8
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Ordinary Differential Equations
• Methods described here are for solving differential equations of the form:
• The methods in this chapter are all one-step methods and have the general format:
where is called an increment function, and is used to extrapolate from an old value yi to a new value yi+1.
dy
dt= f t, y( )
yi+1 = yi +h
Euler’s Method
• The first derivative
provides a direct
estimate of the slope
at ti:
and the Euler method
uses that estimate as
the increment
function:
dy
dt ti
= f ti , yi( )
= f ti , yi( )
yi+1 = yi + f ti , yi( )h
by Lale Yurttas, Texas A&M
University
Chapter 25 12
Figure 25.4
Error Analysis for Euler’s Method
• The numerical solution of ODEs involves two types of error:– Truncation errors, caused by the nature of the techniques employed
– Roundoff errors, caused by the limited numbers of significant digits that can be retained
• The total, or global truncation error can be further split into:– local truncation error that results from an application method in question over a
single step, and
– propagated truncation error that results from the approximations produced during previous steps.
Error Analysis for Euler’s Method
• The local truncation error for Euler’s method is O(h2) and proportional to the derivative of f(t,y) while the global truncation error is O(h).
• This means:
– The global error can be reduced by decreasing the step size, and
– Euler’s method will provide error-free predictions if the underlying function is linear.
• Euler’s method is conditionally stable, depending on the size of h.
Heun’s Method
• One method to improve Euler’s method is to determine derivatives at the beginning and predicted
ending of the interval and average them:
• This process relies on making a prediction of the new value of y, then correcting it based on the
slope calculated at that new value.
• This predictor-corrector approach can be iterated to convergence:
Midpoint Method
• Another improvement to Euler’s method is similar to Heun’s method, but predicts the slope at the midpoint of an interval rather than at the end:
• This method has a local truncation error of O(h3) and global error of O(h2)
Runge-Kutta Methods
• Runge-Kutta (RK) methods achieve the accuracy of a Taylor series approach without requiring the calculation of higher derivatives.
• For RK methods, the increment function can be generally written as:
where the a’s are constants and the k’s are
where the p’s and q’s are constants.
= a1k1 + a2k2 + + ankn
k1 = f ti , yi( )k2 = f ti + p1h, yi + q11k1h( )k3 = f ti + p2h, yi + q21k1h+ q22k2h( )
kn = f ti + pn−1h, yi + qn−1,1k1h+ qn−1,2k2h+ + qn−1,n−1kn−1h( )
Classical Fourth-Order Runge-Kutta Method
• The most popular RK methods are fourth-order, and the most
commonly used form is:
where:
yi+1 = yi +1
6k1 + 2k2 + 2k3 + k4( )h
k1 = f ti , yi( )
k2 = f ti +1
2h, yi +
1
2k1h
k3 = f ti +1
2h, yi +
1
2k2h
k4 = f ti + h, yi + k3h( )
Systems of Equations
• Many practical problems require the solution of a system of equations:
• The solution of such a system requires that n initial conditions be
known at the starting value of t.
dy1
dt= f1 t, y1, y2 , , yn( )
dy2
dt= f2 t, y1, y2 , , yn( )
dyn
dt= fn t, y1, y2 , , yn( )
Figure 20.9
Solution Methods
• Single-equation methods can be used to solve systems of
ODE’s as well; for example, Euler’s method can be used on
systems of equations - the one-step method is applied for
every equation at each step before proceeding to the next
step.
• Fourth-order Runge-Kutta methods can also be used, but
care must be taken in calculating the k’s.
by Lale Yurttas, Texas A&M
University
Chapter 25 22
Figure 25.14
23
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Equações diferenciais parciais
Source:
Chapra, Numerical
Methods for Engineers,
2008
24
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Source: Chapra, Numerical Methods for Engineers, 2008
25
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Linear second-order PDEs are of the form
where A - H are functions of x and y only
Elliptic PDEs: B2 - AC < 0
(steady state heat equations)
Parabolic PDEs: B2 - AC = 0
(heat transfer equations)
Hyperbolic PDEs: B2 - AC > 0
(wave equations)
HGuFuEuCuBuAu yxyyxyxx =+++++ 2
Source:
School of Computing Science, University of Cincinatti,
http://www.cs.uc.edu
26
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Solving PDEs
Finite element method
Finite difference method (our focus)
▪ Converts PDE into matrix equation• Linear system over discrete basis elements
▪ Result is usually a sparse matrix
Finite Volume method
Source:
School of Computing Science, University of Cincinatti,
http://www.cs.uc.edu
27
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Elliptic Equations
Elliptic equations in engineering are typically used to characterizesteady-state, boundary value problems.
For numerical solution of elliptic PDEs, the PDE is transformed intoan algebraic difference equation.
Because of its simplicity and general relevance to most areas ofengineering, we will use a heated plate as an example for solvingelliptic PDEs.
Source: Chapra, Numerical Methods for Engineers, 2008
28
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Source: Chapra, Numerical Methods for Engineers, 2008
29
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
The Laplacian Difference Equations/
04
022
2
2
0
,1,1,,1,1
2
1,,1,
2
,1,,1
2
1,,1,
2
2
2
,1,,1
2
2
2
2
2
2
=−+++
=
=
+−+
+−
+−=
+−=
=
+
−+−+
−+−+
−+
−+
jijijijiji
jijijijijiji
jijiji
jijiji
TTTTT
yx
y
TTT
x
TTT
y
TTT
y
T
x
TTT
x
T
y
T
x
T
Laplacian difference
equation.
Holds for all interior points
Laplace Equation
O[(x)2]
O[(y)2]
Source: Chapra, Numerical Methods for Engineers, 2008
30
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Source: Chapra, Numerical Methods for Engineers, 2008
31
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
In addition, boundary conditions along the edges must be
specified to obtain a unique solution.
The simplest case is where the temperature at the boundary
is set at a fixed value, Dirichlet boundary condition.
A balance for node (1,1) is:
Similar equations can be developed for other interior points
to result a set of simultaneous equations.
04
0
75
04
211211
10
01
1110120121
=++−
=
=
=−+++
TTT
T
T
TTTTT
Source: Chapra, Numerical Methods for Engineers, 2008
32
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
1504
1004
1754
504
04
754
504
04
754
332332
33231322
231312
33322231
2332221221
13221211
323121
22132111
122111
=+−−
=−+−−
=−+−
=−+−−
=−−+−−
=−−+−
=−+−
=−−+−
=−−
TTT
TTTT
TTT
TTTT
TTTTT
TTTT
TTT
TTTT
TTT
• The result is a set of nine simultaneous equations with nine
unknowns:
Source: Chapra, Numerical Methods for Engineers, 2008
33
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Soluções
• Método de eliminação Gauss
• Método de decomposição LU
• Método iterativo (aproximado) de Gauss-Seidel
mais detalhes.pdf
34
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Parabolic Equations
Parabolic equations are employed to characterize time-variable
(unsteady-state) problems.
Conservation of energy can be used to develop an unsteady-state
energy balance for the differential element in a long, thin insulated
rod.
Source: Chapra, Numerical Methods for Engineers, 2008
35
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Energy balance together with Fourier’s law ofheat conduction yields heat-conductionequation:
Just as elliptic PDEs, parabolic equations canbe solved by substituting finite divideddifferences for the partial derivatives.
In contrast to elliptic PDEs, we must nowconsider changes in time as well as in space.
Parabolic PDEs are temporally open-ended andinvolve new issues such as stability.
t
T
x
Tk
=
2
2
Source: Chapra, Numerical Methods for Engineers, 2008
36
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
36
Source: Chapra, Numerical Methods for Engineers, 2008
37
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Explicit Methods
The heat conduction equation requires approximations for the second derivative in space and
the first derivative in time:
( )
( )( )211
1
1
2
11
1
2
11
2
2
2
2
2
x
tkTTTTT
t
TT
x
TTTk
t
TT
t
T
x
TTT
x
T
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
=+−+=
−=
+−
−=
+−=
−+
+
+
−+
+
−+
Source: Chapra, Numerical Methods for Engineers, 2008
This equation can be written for all interior nodes on the rod.
It provides an explicit means to compute values at each node for a future
time based on the present values at the node and its neighbors.
38
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
Source: Chapra, Numerical Methods for Engineers, 2008
39
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.msA simple Implicit Method
Implicit methods overcome difficulties
associated with explicit methods at the
expense of somewhat more complicated
algorithms.
In implicit methods, the spatial derivative is
approximated at an advanced time interval
l+1:
which is second-order accurate.
( )2
1
1
11
1
2
2 2
x
TTT
x
T l
i
l
i
l
i
+−
+
−
++
+
Source: Chapra, Numerical Methods for Engineers, 2008
40
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.tobias.br.ms
( )
( )
( )( ) ( )
( ) ( )1
1
1
1
1
1
01
1
2
1
1
1
0
1
0
1
1
11
1
1
2
1
1
11
1
21
21
21
2
+
+
+
−
+
+++
++
+
+
++
−
++
−
++
+
+=−+
=
+=−+
=
=−++−
−=
+−
l
m
l
m
l
m
l
m
llll
ll
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
l
i
tfTTT
mi
tfTTT
tfT
TTTT
t
TT
x
TTTk
This eqn. applies to all but the first and the last
interior nodes, which must be modified to reflect
the boundary conditions:
Resulting m unknowns and m linear algebraic
equations
Source: Chapra, Numerical Methods for Engineers, 2008
41
Tobias Bleninger
Universidade Federal do Paraná (UFPR)
Departamento de Engenharia Ambiental(DEA)
www.Bleninger.info
Referências
• Chapra, Numerical Methods for Engineers, 2008
• School of Computing Science, University of Cincinatti, http://www.cs.uc.edu
• Markus Uhlmann, "Numerical Fluid Mechanics I", IfH, Karlsruhe Institute of Technology
(www.ifh.kit.edu)
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