incompressible viscous flows via finite difference methods

... 399 402 402 404 12 Incompressible Viscous Flows via Finite Element Methods 12.1 Primitive Variable Methods 12.1.1 Mixed Methods 12.1.2 Penalty Methods 12.1.3 Pressure Correction Methods 12.1.4 Generalized ... FINITE DIFFERENCE METHODS Derivation of Finite Difference Equations 3.1 Simple Methods 3.2 General Methods 3.3 Higher Order Derivatives 45 45 46 50 v vi CONTENTS 3.4 Multidimensional Finite Difference ... Computations by Finite Difference Methods 1.3 One-Dimensional Computations by Finite Element Methods 1.4 One-Dimensional Computations by Finite Volume Methods 1.4.1 FVM via FDM 1.4.2 FVM via FEM 1.5...

Ngày tải lên: 01/04/2014, 10:45

... Solution 3.1 Long-Wave Component Inside Finite Mach Number Region 3.2 Short-Wave Component Inside Finite Mach Number Region 3.3 Fundamental Solution Outside Finite Mach Number Region 15 17 35 39 ... Navier-Stokes Equation and their large-time asymptotic behavior On the other hand, to apply her methods to deal with the question with the presence of boundary, it is necessary to find Green’s ... discuss the structure outside this region at the end of the section 3.1 Long-Wave Component Inside Finite Mach Number Region Denote: c H1 = c F1 = +e ( +i|⇠|)t i|⇠|)t + + +e ( +i|⇠|)t + e( + i|⇠|)t...

Ngày tải lên: 10/09/2015, 09:06

... operation in the above list is used The first row is divided by the element a11 (this being a trivial linear combination of the first row with any other row — zero coefficient for the other row) ... Oxford University Press) [1] Carnahan, B., Luther, H.A., and Wilkes, J.O 1969, Applied Numerical Methods (New York: Wiley), Example 5.2, p 282 Bevington, P.R 1969, Data Reduction and Error Analysis ... matrix, but only halfway, to a matrix whose components on the diagonal and above (say) remain nontrivial Let us now see what advantages accrue Suppose that in doing Gauss-Jordan elimination, as described...

Ngày tải lên: 15/12/2013, 04:15

... we have, via the Cameron-Martin theorem, the infinitesimal change in the Radon-Nikodym derivative of the “shifted” measure with respect to the original Wiener measure This is not trivial since ... cascade down to smaller and smaller scales, until it reaches a “dissipative scale” η at which the viscous term ν∆u dominates the nonlinearity (u · ∇)u in (1.1) This picture is complemented by that ... an infinite number of modes [FM95], [Fer97], [EH01], [MS05] The former used a change of measure via Girsanov’s theorem and the pathwise contractive properties of the dynamics to prove ergodicity...

Ngày tải lên: 29/03/2014, 07:20

... asymptotic behavior of the compressible Navier- Stokes equations for a 1D isothermal viscous gas Math Models Methods Appl Sci 18, 1383–1408 (2008) doi:10.1142/S0218202508003078 17 Zhang, T, Fang, ... Navier-Stokes equations when the initial density is a piece-wise smooth function, having only a finite number of jump discontinuities For the related degenerated density function and viscosity...

Ngày tải lên: 20/06/2014, 22:20

... “Regularity criterion in terms of pressure for the Navier-Stokes equations,” Nonlinear Analysis: Theory, Methods & Applications, vol 46, no 5, pp 727–735, 2001 11 L C Berselli and G P Galdi, “Regularity ... Mathematical Society, vol 130, no 12, pp 3585–3595, 2002 12 Q Chen and Z Zhang, “Regularity criterion via the pressure on weak solutions to the 3D NavierStokes equations,” Proceedings of the American...

Ngày tải lên: 22/06/2014, 02:20

... topic helps to clarify them the key differences between CE and XP You may want to ask if there is any student in the class with this requirement, if no you can abbreviate the topic • Is your device ... features and programming elements API Differences between CE and the Desktop If you are familiar with Win32 programming, you will notice that there are some API differences when programming applications ... strategy to present this module: Selecting a Windows Embedded Operating System Briefly explain the differences between Windows CE NET and Windows XP Embedded and discuss the questions that should...

Ngày tải lên: 18/10/2013, 17:15

... Iterative methods become preferable when the battle against loss of significance is in danger of being lost, either due to large N or because the problem is close to singular We will treat iterative methods ... Chapters 18 and 19 These methods are important, but mostly beyond our scope We will, however, discuss in detail a technique which is on the borderline between direct and iterative methods, namely the ... underlying nature, are close to singular In this case, you might need to resort to sophisticated methods even for the case of N = 10 (though rarely for N = 5) Singular value decomposition (§2.6)...

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15

... are smooth near a root, the methods known respectively as false position (or regula falsi) and secant method generally converge faster than bisection In both of these methods the function is assumed ... previous boundary points is discarded in favor of the latest estimate of the root The only difference between the methods is that secant retains the most recent of the prior estimates (Figure 9.2.1; ... previous uncertainty to the first power (as is the case for bisection), it is said to converge linearly Methods that converge as a higher power, n+1 = constant × ( n )m m>1 (9.1.4) are said to converge...

Ngày tải lên: 15/12/2013, 04:15

... right-side manipulations can be reduced to only N loop executions, and, for matrix inversion, the two methods have identical efficiencies Both Gaussian elimination and Gauss-Jordan elimination share ... LU Decomposition and Its Applications Isaacson, E., and Keller, H.B 1966, Analysis of Numerical Methods (New York: Wiley), §2.1 Johnson, L.W., and Riess, R.D 1982, Numerical Analysis, 2nd ed (Reading, ... successive ones? The advantage is that the solution of a triangular set of equations is quite trivial, as we have already seen in §2.2 (equation 2.2.4) Thus, equation (2.3.4) can be solved by forward...

Ngày tải lên: 24/12/2013, 12:16

... αii ≡ i = 1, , N (2.3.11) A surprising procedure, now, is Crout’s algorithm, which quite trivially solves the set of N + N equations (2.3.8)–(2.3.11) for all the α’s and β’s by just arranging ... as many right-hand sides as we then care to, one at a time This is a distinct advantage over the methods of §2.1 and §2.2 45 2.3 LU Decomposition and Its Applications a c e g i b d di f ag h j ... Gauss-Jordan routine like gaussj (§2.1) to invert a matrix in place, again destroying the original Both methods have practically the same operations count Sample page from NUMERICAL RECIPES IN C: THE...

Ngày tải lên: 24/12/2013, 12:16

... READING: Keller, H.B 1968, Numerical Methods for Two-Point Boundary-Value Problems (Waltham, MA: Blaisdell), p 74 Dahlquist, G., and Bjorck, A 1974, Numerical Methods (Englewood Cliffs, NJ: Prentice-Hall), ... stored compactly, is to multiply it by a vector to its right Although this is algorithmically trivial, you might want to study the following routine carefully, as an example of how to pull nonzero ... tridag"); If this happens then you should rewrite your equations as a set of order N − 1, with u2 trivially eliminated u[1]=r[1]/(bet=b[1]); for (j=2;j

Ngày tải lên: 24/12/2013, 12:16

... This is fun, but let’s look at practicalities: If you estimate how large N has to be before the difference between exponent and exponent log2 = 2.807 is substantial enough to outweigh the bookkeeping...

Ngày tải lên: 24/12/2013, 12:16

... 56 Chapter Solution of Linear Algebraic Equations But (2.5.2) can also be solved, trivially, for δb Substituting this into (2.5.3) gives A · δx = A · (x + δx) − b (2.5.4) #include "nrutil.h" ... (Baltimore: Johns Hopkins University Press), p 74 Dahlquist, G., and Bjorck, A 1974, Numerical Methods (Englewood Cliffs, NJ: Prentice-Hall), §5.5.6, p 183 Forsythe, G.E., and Moler, C.B 1967, ... of SVD in this application to Chapter 15, whose subject is the parametric modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond our scope: Any...

Ngày tải lên: 24/12/2013, 12:16

... N × N say, then U, V, and W are all square matrices of the same size Their inverses are also trivial to compute: U and V are orthogonal, so their inverses are equal to their transposes; W is diagonal, ... very small but nonzero, so that the matrix is ill-conditioned In that case, the direct solution methods of LU decomposition or Gaussian elimination may actually give a formal solution to the set ... can use the decomposition either once or many times with different right-hand sides The crucial difference is the “editing” of the singular values before svbksb is called: Sample page from NUMERICAL...

Ngày tải lên: 24/12/2013, 12:16

... Package methods The advantage of this scheme, which can be called row-indexed sparse storage mode, is that it requires storage of only about two times the number of nonzero matrix elements (Other methods ... System So-called conjugate gradient methods provide a quite general means for solving the N × N linear system A·x =b (2.7.29) The attractiveness of these methods for large sparse systems is that ... convergence of these methods The ordinary conjugate gradient method works well for matrices that are well-conditioned, i.e., “close” to the identity matrix This suggests applying these methods to the...

Ngày tải lên: 24/12/2013, 12:16

... work out its coefficients, and then obtain the numerators and denominators of the specific Pj ’s via synthetic division by the one supernumerary term (See §5.3 for more on synthetic division.) ... require only on the order of N (log N )2 operations, compared to N for Levinson’s method These methods are too complicated to include here Sample page from NUMERICAL RECIPES IN C: THE ART OF ... about When you can use it, Cholesky decomposition is about a factor of two faster than alternative methods for solving linear equations Instead of seeking arbitrary lower and upper triangular factors...

Ngày tải lên: 21/01/2014, 18:20

... work out its coefficients, and then obtain the numerators and denominators of the specific Pj ’s via synthetic division by the one supernumerary term (See §5.3 for more on synthetic division.) ... require only on the order of N (log N )2 operations, compared to N for Levinson’s method These methods are too complicated to include here Sample page from NUMERICAL RECIPES IN C: THE ART OF ... about When you can use it, Cholesky decomposition is about a factor of two faster than alternative methods for solving linear equations Instead of seeking arbitrary lower and upper triangular factors...

Ngày tải lên: 21/01/2014, 18:20

... of Tree Methods 17.2 Efficiency of Simplified Schemes 17.3 Higher Order Markov Chain Approximations 17.4 Finite Difference Methods ... solving the corresponding PIDE via finite difference or finite element methods, see, for instance, D’Halluin, Forsyth & Vetzal (2005) and Cont & Voltchkova (2005) These methods are computationally ... 14.2 Stability of Predictor-Corrector Methods 14.3 Stability of Some Implicit Methods 14.4 Stability of Simplified Schemes...

Ngày tải lên: 19/02/2014, 22:20