solve system of partial differential equations mathematica
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 ... 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 ... What 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...
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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 ... of the
difference equations are so slowly varying as to be considered constant in space
and time. In that case, the independent solutions, or eigenmodes, of the difference
equations are all of ... Colella, P. 1984,
Journal of Computational Physics
, vol. 54, pp. 174–201. [6]
842
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
Ngày tải lên :
15/12/2013, 04:15
... 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 equations. ... auxiliary variables.
The generic problem in ordinary differential equations is thus reduced to the
study of a set of N coupled first-order differential equations for the functions
y
i
,i=1,2, ,N, having ... 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 Press).
16.1...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
Ngày tải lên :
15/12/2013, 04:15
... 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 Press).
16.1 ... 710
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 ... that derive from this basic
712
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 Partial Differential Equations part 3 pptx
Ngày tải lên :
15/12/2013, 04:15
... fully
explicit). To solve equation (19.2.8) one has to solve a set of simultaneous linear
equations at each timestep for the u
n+1
j
. Fortunately, this is a simple problem because
the system is tridiagonal: ... (19.2.22) with n → n +1leaves
us with a nasty set of coupled nonlinear equations to solve at each timestep. Often
there is an easier way: If the form of D(u) allows us to integrate
dz = D(u)du (19.2.23)
analytically ... amplitudes, so that the 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...
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Tài liệu Partial Differential Equations part 4 ppt
Ngày tải lên :
15/12/2013, 04:15
... 856
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... δ
2
y
u
n+1
j,l
(19.3.16)
The advantage of this method is that each substep requires only the solution of a
simple tridiagonal system.
Operator Splitting Methods Generally
The basic idea of operator splitting, ... ∆t)
···
u
n+1
= U
m
(u
n+(m−1)/m
, ∆t)
(19.3.20)
854
Chapter 19. Partial 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 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 ... way to solve equations of the form (19.4.28), including the constant
coefficient problem(19.0.3),isacombinationofFourieranalysisandcyclicreduction,
the FACR method
[3-6]
.Ifattherth stage of CR ... another way of thinking about relaxation methods that is somewhat
more physical. Suppose we wish to solve the elliptic equation
Lu = ρ (19.5.1)
858
Chapter 19. Partial Differential Equations
Sample...
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Tài liệu Partial Differential Equations part 6 doc
Ngày tải lên :
15/12/2013, 04:15
... 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 directly. For production solution of ... Givenu
n
,onesolves(19.5.36) foru
n+1/2
,substitutes
on the right-hand side of (19.5.37), and then solves for u
n+1
. The key question
is how to choose the iteration parameter r, the analog of a choice of ... These
methods can solve elliptic PDEs discretized on 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...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
Ngày tải lên :
24/12/2013, 12:16
... generally useful stepper
routine is this: One of the arguments of the routine will of course be the vector of
dependent variables at the beginning of a proposed step. Call that y[1 n].Let
us require ... Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall). [1]
Cash, J.R., and Karp, A.H. 1990,
ACM Transactions on Mathematical Software
, vol. 16, pp. 201–
222. ... ,n−1
y(x+H)≈y
n
≡
1
2
[z
n
+z
n−1
+hf(x + 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...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
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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 ... Initial Value Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall). [1]
Cash, J.R., and Karp, A.H. 1990,
ACM Transactions on Mathematical Software
, vol. 16, pp. 201–
222. ... section.
The usefulness ofthe modified midpointmethod tothe Bulirsch-Stoertechnique
(§16.4) derives from a “deep” result about equations (16.3.2), due to Gragg. It turns
out that the error of (16.3.2), expressed...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
Ngày tải lên :
24/12/2013, 12:16
... 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 ... method
a degree of robustness for problems with discontinuities.
Let us remind you once again that scaling of the variables is often crucial for
successful integration of differential equations. The ... ordinary differential equations with minimal computational effort. (A possible
exception, infrequently encountered in practice, is discussed in §16.7.)
726
Chapter 16. Integration of Ordinary Differential...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
Ngày tải lên :
24/12/2013, 12:16
... Second-Order Conservative Equations
Usually when you have a system of high-order differential equations to solve it is best
to reformulate them as a system of first-order equations, as discussed ... vol. 27, pp. 505–535.
16.6 Stiff Sets of Equations
As soon as one deals with more than one first-order differential equation, the
possibility of a stiff set of equations arises. Stiffness occurs ... compatibility with bsstep 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...
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
Ngày tải lên :
24/12/2013, 12:16
... compatibility with bsstep 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 ... vol. 27, pp. 505–535.
16.6 Stiff Sets of Equations
As soon as one deals with more than one first-order differential equation, the
possibility of a stiff set of equations arises. Stiffness occurs ... nice feature of implicit methods holds only
for linear systems, but even in the general case implicit methods give better stability.
742
Chapter 16. Integration of Ordinary Differential Equations
Sample...
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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
Ngày tải lên :
24/12/2013, 12:16
... Problems in Ordinary Differential Equations
(Englewood
Cliffs, NJ: Prentice-Hall), Chapter 9. [1]
Shampine, L.F., and Gordon, M.K. 1975,
Computer Solution of Ordinary Differential Equations.
The Initial ... adjusting
the stepsize is difficult.
748
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 ... otherwise
it is implicit. The order of the method depends on how many previous steps we
use to get each new value of y.
Consider how we might solve an implicitformula of the form (16.7.2) for y
n+1
.
Two...
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Tài liệu Partial Differential Equations part 7 doc
Ngày tải lên :
24/12/2013, 12:16
... 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 directly. For production solution of ... accelerate the convergence of the smooth components
of the fine-grid residuals.
880
Chapter 19. Partial Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... These
methods can solve elliptic PDEs discretized on 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...
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