Appendix C: Recapitulation of Numerical schemes

Transcription

1 Appendix C: Recapitulation of Numerical schemes August 31, 2009) SUMMARY: Certain numerical schemes of general use are regrouped here in order to facilitate implementations of simple models C1 The tridiagonal system solver As a special case of the general LU decomposition eg, Riley et al 1997), an efficient tridiagonal system solver, based on the so-called Thomas algorithm, can be constructed We begin by assuming that there exists a decomposition for which the lower L) and upper U) matrices possess the same bandwidth of 2: ¼ ¼ β β β m β m 1 a 1 c b 2 a 2 c b 3 a 3 c b m 1 a m 1 c m b m a m ½¼ ½ α 1 γ α 2 γ α 3 γ α m 1 γ m α m = ½ C1) where the first matrix is the original tridiagonal matrix to be decomposed ie, elements a 1 etc are known), the second matrix is L with one line of non-zero elements below the diagonal, and the last matrix is U with one line of non-zero elements above the diagonal Performing the product of matrices, we identify elements k, k 1), k, k) and k, k + 1) of the product as 723

2 724 APPENDIX C: RECAPITULATION OF NUMERICAL SCHEMES b k = β k α k 1 C2) a k = β k γ k 1 + α k C3) c k = γ k C4) These relations can be solved for the components of L and U by observing that γ k = c k for all k Since the first row demands a 1 = α 1, subsequent rows provide α k and β k recursively from β k = b k α k 1, α k = a k β k c k 1, k = 2,, m C5) provided that no α k is zero otherwise the matrix is singular and cannot be decomposed) Note that there is no β 1 The tridiagonal matrix A has now been decomposed as the product of lower and upper triangular matrices The solution of Ax = LUx = f is then obtained by first solving Ly = f β β β m β m 1 y 1 y 2 y 3 y m 1 y m = by proceeding from the first row downward and then solving Ux = y α 1 γ x 1 0 α 2 γ x α 3 γ 3 0 x 3 = α m 1 γ m 1 x m α m x m by proceeding from the bottom row upward The solution is f 1 f 2 f 3 f m 1 f m y 1 y 2 y 3 y m 1 y m C6), C7) y 1 = f 1, y k = f k β k y k 1, k = 2,, m C8) x m = y m α m, x k = y k γ k x k+1 α k, k = m 1,, 1 C9) In practice, the α values can be stored in vector a initially holding the values a k, since once α k is known, a k is not needed anymore Similarly β values can be stored in vector b initially holding b k and γ values in a vector c Also the y and f values can share the same vector f as the f k value is no longer needed once the y k value has been computed With the additional vector x for the solution, only 5 vectors are required, and the solution is obtained with only three loops over m points This demands approximately 5m floating-point operations instead of the m 3 operations that a brutal matrix inversion would have required The algorithm is implemented in MATLAB file thomasm

3 C2 1D finite-difference schemes of various orders 725

4 726 APPENDIX C: RECAPITULATION OF NUMERICAL SCHEMES Table C1 STANDARD FINITE-DIFFERENCE OPERATORS FOR UNIFORM GRIDS FORWARD DIFFERENCE O t) u n u n+1 u n+2 u n+3 u n+4 FORWARD DIFFERENCE O t 2 ) u n u n+1 u n+2 u n+3 u n+4 u n+5 t u t -1 1 t 2 2 u t t 3 3 u t t 4 4 u t t u t t 2 2 u t t 3 3 u t t 4 4 u t BACKWARD DIFFERENCE O t) u n 4 u n 3 u n 2 u n 1 u n t u t -1 1 t 2 2 u t t 3 3 u t t 4 4 u t BACKWARD DIFFERENCE O t 2 ) u n 5 u n 4 u n 3 u n 2 u n 1 u n 2 t u t t 2 2 u t t 3 3 u t t 4 4 u t CENTRAL DIFFERENCE O t 2 ) u n 2 u n 1 u n u n+1 u n+2 2 t u t t 2 2 u t t 3 3 u t t 4 4 u t CENTRAL DIFFERENCE O t 4 ) u n 3 u n 2 u n 1 u n u n+1 u n+2 u n+3 12 t u t t 2 2 u t t 3 3 u t t 4 4 u t

5 C3 Time-stepping algorithms 727

6 728 APPENDIX C: RECAPITULATION OF NUMERICAL SCHEMES Table C2 STANDARD TIME-STEPPING METHODS FOR du/dt = Qt, u) EULER METHODS Scheme Order Explicit ũ n+1 = ũ n + tq n t Implicit ũ n+1 = ũ n + tq n+1 t Trapezoidal ũ n+1 = ũ n + t 2 Q n + Q n+1) t 2 General ũ n+1 = ũ n + t 1 α)q n + αq n+1) t Runge- Kutta Runge- Kutta MULTI-STAGE METHODS Scheme Order ũ n+1/2 = ũ n + t 2 Qtn, ũ n ) ũ n+1 = ũ n + tqt n+1/2, ũ n+1/2 ) t 2 ũa n+1/2 = ũ n + t 2 Qtn, ũ n ) ũ n+1/2 b = ũ n + t 2 Qtn+1/2, ũa n+1/2 ) ũ = ũ n + t Qt n+1/2, ũ n+1/2 b ) ũ n+1 = ũ n 1 + t 6 Qtn, ũ n ) Qtn+1/2, ũa n+1/2 ) ) Qtn+1/2, ũ n+1/2 b ) Qtn+1, ũ ) t 4 MULTI-STEP METHODS Scheme Truncation order Leapfrog ũ n+1 = ũ n tq n t 2 Adams- Bashforth ũ n+1 = ũ n + t 2 Q n 1 + 3Q n) t 2 Adams- Moulton ũ n+1 = ũ n + t 12 Q n 1 + 8Q n + 5Q n+1) t 3 Adams- Bashforth ũ n+1 = ũ n + t 12 5Q n 2 16Q n Q n+1) t 3 PREDICTOR-CORRECTOR METHODS Scheme Order ũ = ũ n + tqt n, ũ n ) Heun ũ n+1 = ũ n + t 2 Qt n, ũ n ) + Qt n+1, ũ ) ) t 2 Leapfrog- ũ = ũ n tq n Trapezoidal ũ n+1 = ũ n + t 2 Q n + 5Qt n+1, ũ ) ) t 2 ũ = ũ n + t 2 Q n 1 + 3Q n) ABM ũ n+1 = ũ n + t 12 Q n 1 + 8Q n + 5Qt n+1, ũ ) ) t 3

7 729 C4 Partial-derivatives finite differences On a regular grid x = i x + x 0, y = j y + y 0, the following expressions are of second order Jacobian Ja, b) = a b x y b a x y J ++ i,j = a i+1,j a i 1,j )b i,j+1 b i,j 1 ) b i+1,j b i 1,j )a i,j+1 a i,j 1 ) 4 x y J + i,j = [a i+1,jb i+1,j+1 b i+1,j 1 ) a i 1,j b i 1,j+1 b i 1,j 1 )] 4 x y [a i,j+1b i+1,j+1 b i 1,j+1 ) a i,j 1 b i+1,j 1 b i 1,j 1 )] 4 x y J + i,j = [b i,j+1a i+1,j+1 a i 1,j+1 ) b i,j 1 a i+1,j 1 a i 1,j 1 )] 4 x y [b i+1,ja i+1,j+1 a i+1,j 1 ) b i 1,j a i 1,j+1 a i 1,j 1 )] 4 x y Cross derivatives 2 u x y u i+1,j+1 u i+1,j + u i,j u i,j+1 i+1/2,j+1/2 x y 2 u x y u i+1,j+1 u i+1,j 1 + u i 1,j 1 u i 1,j+1 i,j 4 x y Laplacian 2 u x u y 2 u i+1,j + u i 1,j 2u i,j i,j x 2 + u i,j+1 + u i,j 1 2u i,j y 2 C5 DFT and FFT u 0 L x x Figure C-1 Periodic signal sampled with N evenly-spaced points x 0 to x Note that no x N is needed since by periodicity the value of the function in x N is the same as in x 0

8 730 APPENDIX C: RECAPITULATION OF NUMERICAL SCHEMES In a periodic domain in which x varies between 0 and L, a complex function ux) can be expanded in Fourier modes according to ux) = + n= a n e i n 2πx L C10) Using orthogonality properties of the Fourier modes 9 1 L e i n m)2πx L dx = δnm, C11) L 0 the complex coefficients a n are readily obtained by multiplying C10) by exp i m2πx/l) and integrating a m = 1 L L 0 2πx i m ux)e L dx C12) Note that we could have used a n = b n /L where b n would then be calulated using the integral without the normalization 1/L Discrete Fourier Transform DFT) simply truncates 10 the infinite series to a finite series ũx) = a n e i n 2πx L C13) It uses ux) at a series of sampling points the discrete values on a numerical grid for example) to determine the coefficients a n The points are regularly spaced x j = j x, j = 0,, N 1 Because of periodicity, using x N would amount to use x 0 and the point is therefore not retained in the sampling Complex coefficients can be evaluated as a n = 1 N = 1 N j=0 j=0 ux j )e i n 2πx j L u j e i nj 2π N C14) where we write ux j ) = u j to highlight the fact that we only use information at the discrete points Obviously this is a discrete version of the integral C12) The interesting point is that coefficients a m are exact in the sense that if we evaluate ũx j ) with those coefficients, we obtain the values at the grid points ux j ) The proof of this result is not trivial and we must verify that if we use coefficients C14) 9 δ ij stands for the Kronecker symbol 0 if i j, 1 otherwise) 10 The truncation from 0 to N 1 is somehow arbitrary and sometimes the presentation with a series from N/2 to N/2 is preferred

9 731 in C13) we obtain ũx j ) = ux j ) at the grid points: We obtain successively where ũx j ) = = = 1 N = 1 N a n e i n 2πx j L 1 N m=0 m=0 m=0 u m e i nm 2π N e i nj 2π N [ u m [ u m ρ = e i j m) 2π N e i j m)n 2π N ρ n ] For j m, ρ 1 and the geometric sum takes the value ] C15) C16) C17) ρ n = 1 ρn 1 ρ = 0 C18) because j m is an integer so that ρ N = 1 When j = m the sum between brackets is simply N and we arrive at ũx j ) = ux j ), the desired result Transformation C14) is called the direct transform DFT) and C15) the inverse transform IDFT) As for the Fourier series, scaling can vary and the factor 1/N is commonly applied in the inverse transform rather than the direct Note that the inverse and direct transform are almost identical transformations, except for this scaling and more importantly the sign in the exponential Once the coefficients known, C13) can be evaluated at any position x in the domain and because the function ũx) takes the values ux j ) on the discrete points, we now have an interpolation function It can be used to calculate the value of the function at any desired location using N discrete values Also C13) can be used to evaluate derivatives: dũ dx = i k n a n e i knx, k n = 2πn L C19) All we have to do to calculate derivatives is to create Fourier coefficients b n = i k n a n from the original coefficient These are the Fourier coefficient for the derivative, which, by way of an inverse transform, lead to the values of derivatives ux) sampling ux j ) DFTu j) a n derivation b n = i k n a n IDFTb n ) dũ dx We note that the wavenumber associated with N m in C13) is equivalent to the one associated with m In reality, the shortest signal resolved by the series corresponds to n = N/2 with wavenumber k = Nπ/L or wavelength 2 x Hence coefficient a 0 contains the information on the average value, a 1 on the wavelength L and so on up to a N/2 which contains the xj

10 732 APPENDIX C: RECAPITULATION OF NUMERICAL SCHEMES Fundamental mode Average value 0 a n 2 x mode N/2 N 1 n Figure C-2 Coefficients a n contain the amplitudes of the different modes With the summation chosen from 0 to N 1, a N/2 contains the amplitude of the shortest signal complex amplitude of the shortest signal We see why some presentations of the DFT define the series between N/2 and N/2 The information content is however the same Up to now, the functions treated were considered complex functions and for a real signal u there is a redundancy in the coefficients a n Indeed, with u j = u j, where u stands for the complex conjugate, we have a N m = 1 N = 1 N j=0 j=0 u j e i N m)j 2π N u j e +i mj 2π N = a n C20) For real functions u, the DFT contains thus some redundant information and some algorithms exploit this property Cosine transforms CFT and sine transforms SFT perform in a similar way as DFT using only cosine or sine functions They are particularly useful if the solution of a problem is known to satisfy homogeneous boundary conditions For Neuman conditions in x = 0 and x = L, CFT is the method of choice while for homogeneous Dirichlet conditions, the development of the solution in terms of sine functions allows to satisfy these boundary conditions easily The annoying aspect with the transformation presented up to now is that a straightforward implementation as a sum of factors needs a data array of length N, and for each coefficient a n the calculation of N terms Therefore, in total, N 2 operations are necessary for each transform This is prohibitively expensive when used in large-size problems Fortunately Cooley and Tukey introduced in 1965 a method which is probably among the most celebrated numerical algorithms, reducing the cost of the DFT to N log 2 N operations Interestingly enough, the original idea of the method, now called Fast Fourier Transform, goes back to Gauss in 1805 see collected reprints Gauss, 1866), long before the advent of calculators able to exploit it The Fast Fourier Transform is a practical calculation of the discrete Fourier transform and starts with the observation that we can write the transform by separating even and odd terms in j:

11 733 Na n = = j = 0 j even N/2 1 m=0 u j e i nj 2π N + j = 0 j odd u j e i nj 2π N 2π i nm u 2m e N/2) + e i n 2π N N/2 1 m=0 2π i nm u 2m+1 e N/2) C21) and for simplicity we assume N to be a power of two, which is also the case in which the method performs best Each of the two sums involved is nothing else than a discrete transform working on N/2 data points But those in turn can be broken up into two smaller pieces and so on, a technique called divide and conquer When N is a power of two, this recursive division is possible until all series contain only a single term Such a single data point can be transformed in a single operation and the recurrence can be rolled up back to retrieve the transform of the original series To estimate the cost CN) for a transformation with N data, we see that it is the cost of two transformations of size N/2 and the multiplication by e i n 2π N for each of the N coefficients a n Hence CN) = 2 C N/2) + N C22) For a single point, one operation is needed so that we can deduce CN) N log 2 N C23) There is thus a substantial gain compared to a brute-force approach, which is particularly interesting in spectral methods Section 184) where direct and inverse transforms are intensively used Interpolation of a function knowing expansion coefficients could be obtained by a bruteforce summation of C13) at the desired locations But we can do better Suppose we want to interpolate into a regular finer) grid: x k = k/ml, k = 0,, M 1 having at our disposal measurements from original data on another regular grid x j = j/nl, j = 0,, N 1 The interpolation can be performed very efficiently by simply assuming that shorter signal than those initially capture by the N points have zero amplitude In other words, from a perspective of the new grid of M points, we take the Fourier coefficients provided from the N points up to the shortest wavelength resolved by this grid) and set all coefficients corresponding to shorter wavelength to zero This amounts to add zeros to the array of Fourier coefficients, a procedure called padding With the summing convention taken here from 0 to N 1), the padding must add zeros in the the middle of the array a n Depending on the scaling used, the coefficients also can need a rescaling before transforming back to real space The interpolation procedure is therefore schematically described by: ux j ) FFTu j) a n Padding b n = a n plus zeros; scaling IFFTb n) ũx k )

12 734 APPENDIX C: RECAPITULATION OF NUMERICAL SCHEMES Analytical Problems C-1 Find the truncation errors of the two Adams-Bashforth schemes by a Taylor expansion C-2 Which of the two second-order approximations of the cross-derivative has a lower truncation error? C-3 Try to establish a finite-difference approximation of a Jacobian at a corner point i + 1/2, j + 1/2, using only values from the nearest grid points Numerical Exercises C-1 Compare the behavior of the second-order Adams-Bashforths method with the leapfrogtrapezoidal method on the calculation of an inertial oscillation C-2 Discretize a function ux, y) = sin2πx/l) cos2πy/l) on a regular grid in x, y) Calculate the numerical Jacobian between ũ and ũ, ũ and ũ 2, ũ and ũ 3, and interpret your result C-3 Perform a FFT on fx) = sin2πx/l) between x = 0 and x = L, by sampling with 10, 20 or 40 points Using the spectral coefficients from the FFT, plot the series expansion using a very fine resolution 200 points) and verify that you recover the initial function Then repeat with the function fx) = x What do you observe? C-4 Redo Exercise C-3 using the padding technique instead of a brute-force series evaluation to plot the expansion