applications of partial differential equations in civil engineering

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... 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...

Ngày tải lên: 28/10/2013, 22:15

... 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). ... stability of a differencing scheme. The solution of a hyperbolic problem at a point depends on information within some domain of dependency to the past, shown here shaded. The differencing scheme...

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... 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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... 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 ... 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 (C)...

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...

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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... 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...

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...

Ngày tải lên: 15/12/2013, 04:15

... 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)...

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 ... step h instead of the two required by second-order Runge-Kutta. Perhaps there are applications where the simplicity of (16.3.2), easily coded in- line in some other program, recommends it. In general,...

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...

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....

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; ... methods have been, we think, squeezed 740 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...

Ngày tải lên: 24/12/2013, 12:16