morton mayers numerical solution of partial differential equations pdf
Partial Differential Equations part 1
... 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 ... conceptually, to the solution of large numbers of simultaneous algebraic equations When such equations are nonlinear, they are usually solved by linearization and iteration; so without much loss of generality ... problem as being the solution of special, large linear sets of equations As an example, one which we will refer to in §§19.4–19.6 as our “model problem,” let us consider the solution of equation (19.0.3)...
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Partial Differential Equations part 2
... coefficients 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 ... j 838 Chapter 19 Partial Differential Equations stable unstable ∆t ∆t ∆x ∆x x or j (a) ( b) Figure 19.1.3 Courant condition for stability of a differencing scheme The solution of a hyperbolic ... rewritten as 840 Chapter 19 Partial Differential Equations Other Varieties of Error ξ = e−ik∆x + i − v∆t ∆x sin k∆x (19.1.25) An arbitrary initial wave packet is a superposition of modes with different...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
... subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 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) ... 708 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 2 pptx
... sequence of steps in identical manner Prior behavior of a solution is not used in its propagation This is mathematically proper, since any point along the trajectory of an ordinary differential ... is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations You input the values of the independent variables, and you get out new values which are stepped ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...
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Tài liệu Partial Differential Equations part 3 pptx
... 19 Partial Differential Equations The physical interpretation of the restriction (19.2.6) is that the maximum allowed timestep is, up to a numerical factor, the diffusion time across a cell of ... Ames, W.F 1977, Numerical Methods for Partial Differential Equations, 2nd ed (New York: Academic Press), Chapter Goldberg, A., Schey, H.M., and Schwartz, J.L 1967, American Journal of Physics, vol ... 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
... These will occupy us for the remainder of the chapter CITED REFERENCES AND FURTHER READING: Ames, W.F 1977, Numerical Methods for Partial Differential Equations, 2nd ed (New York: Academic Press) ... PDEs When you increase the order of a differencing method to greater than the order of the original PDEs, you introduce spurious solutions to the difference equations This does not create a problem ... αx sin ky ∆)2 856 Chapter 19 Partial Differential Equations (19.3.13) Called the alternating-direction implicit method (ADI), this embodies the powerful concept of operator splitting or time...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
... effectively fewer equations than unknowns In this case there can be either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution ... combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding the solution space of A involves • Singular value decomposition of a matrix ... written as the N ×N set of equations (AT · A) · x = (AT · b) (2.0.4) where AT denotes the transpose of the matrix A Equations (2.0.4) are called the normal equations of the linear least-squares...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
... two rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Ralston, A., and Rabinowitz, P 1978, A First Course in Numerical Analysis, 2nd ed (New ... (outside North America) a11 a21 a31 a41 38 Chapter Solution of Linear Algebraic Equations Pivoting Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
... applications only the part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) ... purposes, because of its greater diagnostic capability in pathological cases.) Updating a QR decomposition Some numerical algorithms involve solving a succession of linear systems each of which differs ... Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a matrix factorization in O(N ) operations and use the new factorization to solve the next set of...
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Tài liệu Partial Differential Equations part 5 ppt
... 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 ... North America) ∂u = g(y) ∂x 862 Chapter 19 Partial Differential Equations The finite-difference form of equation (19.4.28) can be written as a set of vector equations uj−1 + T · uj + uj+1 = gj ∆2 ... somewhat more general; its applicability is related to the question of whether the equations are separable (in the sense of “separation of variables”) Both methods require the boundaries to coincide...
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Tài liệu Partial Differential Equations part 6 doc
... 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 large ... Chapter 19 Partial Differential Equations ADI (Alternating-Direction Implicit) Method The ADI method of §19.3 for diffusion equations can be turned into a relaxation method for elliptic equations ... left-hand sides of equations (19.5.36) and (19.5.37) are tridiagonal (and usually positive definite), so the equations can be solved by the Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... also occur to you that, ignoring terms of order h6 and higher, we can solve the two equations in (16.2.1) to improve our numerical estimate of the true solution y(x + 2h), namely, y(x + 2h) = ... 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 ... steps (see Figure 16.2.1) How much overhead is this, say in terms of the number of evaluations of the right-hand sides? Each of the three separate Runge-Kutta steps in the procedure requires...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... powers of h, 724 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... method does an excellent job of feeling its way through rocky or discontinuous terrain It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations A second warning ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
... 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 ... 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 ... 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...
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