uses of partial differential equations in engineering
Partial Differential Equations part 1
Ngày tải lên :
28/10/2013, 22:15
... 19. Partial Differential
Equations
19.0 Introduction
The numerical treatment of partial differential equations is, by itself, a vast
subject. Partial differential equations are at the heart of ... are the variables?
• What equations are satisfied in the interior of the region of interest?
• What equations are satisfied by points on the boundary of the region of
interest? (Here Dirichlet ... entiresecond
volume of Numerical Recipes dealing with partial differential equations alone. (The
references
[1-4]
provide, of course, available alternatives.)
In most mathematics books, partial differential equations...
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Partial Differential Equations part 2
Ngày tải lên :
07/11/2013, 19:15
... various ways of improving the accuracy of first-order upwind
differencing. In the continuum equation, material originally a distance v∆t away
840
Chapter 19. Partial Differential Equations
Sample ... own domain of dependency determined by the choice
of points on one time slice (shown as connected solid dots) whose values are used in determining a new
point (shown connected by dashed lines). ... viscosity to the equations, modeling the way Nature uses real viscosity
to smooth discontinuities. A good starting point for trying out this method is the
differencing scheme in §12.11 of
[1]
. This...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
Ngày tải lên :
15/12/2013, 04:15
... America).
Chapter 16. Integration of Ordinary
Differential Equations
16.0 Introduction
Problems involving ordinary differential equations (ODEs) can always be
reduced to the study of sets of first-order differential ... description of each of these types follows.
1. Runge-Kutta methods propagate a solution over an interval by combining
the information from several Euler-style steps (each involving one evaluation of the
right-hand ... 1973,
Computational Methods in Ordinary Differential Equations
(New York: Wiley).
Lapidus, L., and Seinfeld, J. 1971,
Numerical Solution of Ordinary Differential Equations
(New
York: Academic...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
Ngày tải lên :
15/12/2013, 04:15
... discussion of the pitfalls in
constructing a good Runge-Kutta code is given in
[3]
.
Here is the routine for carrying out one classical Runge-Kutta step on a set
of n differential equations. You input ... 1973,
Computational Methods in Ordinary Differential Equations
(New York: Wiley).
Lapidus, L., and Seinfeld, J. 1971,
Numerical Solution of Ordinary Differential Equations
(New
York: Academic ... the solution through an interval h, but uses derivative information only
at the beginning of that interval (see Figure 16.1.1). That means (and you can verify
by expansion in power series) that...
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Tài liệu Partial Differential Equations part 3 pptx
Ngày tải lên :
15/12/2013, 04:15
... evolution of the larger-scale features
of interest takes place superposed with a kind of “frozen in (though fluctuating)
background of small-scale stuff. This answer gives a differencing scheme ... form again and in practice usually retains
the stability advantages of fully implicit differencing.
Schr
¨
odinger Equation
Sometimes the physical problem being solved imposes constraints on ... evolve through of order λ
2
/(∆x)
2
steps before things start to happen on the
scale of interest. This number of steps is usually prohibitive. We must therefore
find a stable way of taking timesteps...
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Tài liệu Partial Differential Equations part 4 ppt
Ngày tải lên :
15/12/2013, 04:15
... underlying PDEs, perhaps allowing second-order
spatial differencing for first-order -in- space PDEs. When you increase the order of
a differencing method to greater than the order of the original ... Recipes Software.
Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-
readable files (including this ... America).
are using is known to be extremely stable, we do not recommend anything higher
than second-order in time (for sets of first-order equations) . For spatial differencing,
we recommend the order of the...
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Tài liệu Partial Differential Equations part 5 ppt
Ngày tải lên :
15/12/2013, 04:15
... level of CR, we have reduced the number of equations by a factor of
two. Since the resulting equations are of the same form as the original equation, we
can repeat the process. Taking the number of ... y-values on these
x-lines. Then fill in the intermediate x-lines as in the original CR algorithm.
The trick is to choose the number of levels of CR so as to minimize the total
number of arithmetic operations. ... mentioned in §19.0, relaxation methods involve splitting the sparse
matrix that arises from finite differencing and then iterating until a solution is found.
There is another way of thinking about...
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Tài liệu Partial Differential Equations part 6 doc
Ngày tải lên :
15/12/2013, 04:15
... y-values on these
x-lines. Then fill in the intermediate x-lines as in the original CR algorithm.
The trick is to choose the number of levels of CR so as to minimize the total
number of arithmetic operations. ... N grid points in O(N ) operations.
The “rapid” direct elliptic solvers discussed in §19.4 solve special kinds of elliptic
equations in O(N log N) operations. The numerical coefficients in these ... the version of
SOR implemented below, we shall adopt odd-even ordering.
The last practical point is that in practice the asymptotic rate of convergence
in SOR is not attained until of order J...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
Ngày tải lên :
24/12/2013, 12:16
... methods.
free_vector(ytemp,1,n);
free_vector(ak6,1,n);
free_vector(ak5,1,n);
free_vector(ak4,1,n);
free_vector(ak3,1,n);
free_vector(ak2,1,n);
}
Noting that the above routines are all in single precision, don’t be too greedy in
specifying eps. Thepunishment forexcessive greediness is interestingand worthyof
Gilbertand Sullivan’sMikado: ... North America).
including garden-variety ODEs or sets of ODEs, and definite integrals (augmenting
the methods of Chapter 4). For storage of intermediate results (if you desire to
inspect them) we ... + H, z
n
)]
(16.3.2)
714
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C)...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
Ngày tải lên :
24/12/2013, 12:16
... 722
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... hmin) nrerror("Step size too small in odeint");
h=hnext;
}
nrerror("Too many steps in routine odeint");
}
CITED REFERENCES AND FURTHER READING:
Gear, C.W. 1971,
Numerical Initial ... Recipes Software.
Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-
readable files (including this...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
Ngày tải lên :
24/12/2013, 12:16
... remind you once again that scaling of the variables is often crucial for
successful integration of differential equations. The scaling “trick” suggested in
the discussion following equation (16.2.8) ... extrapolate each
component of a vector of quantities.
728
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright ... encountered in practice, is discussed in §16.7.)
726
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
Ngày tải lên :
24/12/2013, 12:16
... is
a particular class of equations that occurs quite frequently in practice where you can gain
about a factor of two in efficiency by differencing the equations directly. The equations are
second-order ... 734
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... the arrays y and d2y are of length 2n for a
system of n second-order equations. The values of y arestoredinthefirstnelements of y,
while the first derivatives are stored in the second n elements....
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
Ngày tải lên :
24/12/2013, 12:16
... as
in the original Bulirsch-Stoer method.
The starting point is an implicit form of the midpoint rule:
y
n+1
− y
n−1
=2hf
y
n+1
+ y
n−1
2
(16.6.29)
738
Chapter 16. Integration of Ordinary Differential ... calculates
dydx
.
{
void lubksb(float **a, int n, int *indx, float b[]);
void ludcmp(float **a, int n, int *indx, float *d);
int i,j,nn,*indx;
float d,h,x,**a,*del,*ytemp;
indx=ivector(1,n);
a=matrix(1,n,1,n);
del=vector(1,n);
ytemp=vector(1,n);
h=htot/nstep; ... components of
y
.
{
void jacobn(float x, float y[], float dfdx[], float **dfdy, int n);
void lubksb(float **a, int n, int *indx, float b[]);
void ludcmp(float **a, int n, int *indx, float *d);
int i,j,jtry,*indx;
float...
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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
Ngày tải lên :
24/12/2013, 12:16
... part of the procedure, and it is desirable to minimize its effect. Therefore, the
integration steps of a predictor-corrector method are overlapping, each one involving
several stepsize intervals ... method .In
functional iteration, we take some initial guess for y
n+1
, insert it into the right-hand
side of (16.7.2) to get an updated value of y
n+1
, insert this updated value back into
the ... been, we think, squeezed
752
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C)...
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Tài liệu Partial Differential Equations part 7 doc
Ngày tải lên :
24/12/2013, 12:16
... solving a larger
problem once only, where ease of programming outweighs expense of computer
time. Occasionally, the sparse matrix methods of §2.7 are useful for solving a set
of difference equations ... N grid points in O(N) operations.
The “rapid” direct elliptic solvers discussed in §19.4 solve special kinds of elliptic
equations in O(N log N) operations. The numerical coefficients in these ... the order of the
interpolation P (i.e., it interpolates polynomials of degree m
p
− 1 exactly). Suppose
m
r
is the order of R,andthatRis the adjoint of some P (not necessarily the P you
intend to...
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