solve system of partial differential equations mathematica

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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 ... 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...

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 ... 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...

Ngày tải lên: 07/11/2013, 19:15

... 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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... 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...

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...

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)...

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...

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...

Ngày tải lên: 15/12/2013, 04:15

... 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...

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 ... 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...

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...

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...

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...

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...

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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