2. Introduction
• “The profound study of nature is the most
fertile source of mathematical discoveries.”
---------------Joseph Fourier
3. Definition of the Fourier Transform
and Examples
• The Fourier transform of f(x) is denoted by F{f(x)} = F(k), k ∈ R, and defined
by the integral
where F is called the Fourier transform operator or the Fourier
transformation.
The inverse Fourier transform, denoted by 𝐹−1{F(k)} = f(x), is defined by
where 𝐹−1 is called the inverse Fourier transform operator.
10. Convolution Therorem
• The convolution of two integrable functions f(x) and g(x), denoted by (f ∗
g)(x), is defined by
• provided the integral in (2.5.10) exists, where the factor
1
2𝜋
is a matter of
choice.
13. The convolution has the following algebraic
properties:
• f ∗ g = g ∗ f (Commutative), (2.5.14)
• f ∗ (g ∗ h)=(f ∗ g) ∗ h (Associative), (2.5.15)
• (αf + βg) ∗ h = α (f ∗ h) + β (g ∗ h)
(Distributive), (2.5.16)
• f ∗ 2𝜋𝛿= f = 2𝜋𝛿∗ f (Identity), (2.5.17)
where α and β are constants.
25. Uses
Many linear boundary value and initial value problems in
applied mathematics, mathematical physics, and
engineering science can be effectively solved by the use of
the Fourier transform, the Fourier cosine transform, or the
Fourier sine transform. These transforms are very useful for
solving differential or integral equations.
the replacement of the continuous derivatives in the governing partial differential equations with equivalent finite difference expressions and the rearrangement of the resulting algebraic equation into an algorithm.
In practice the algebraic equations that result from the discretisation process, Sect. 3.1, are obtained on a finite grid. It is to be expected, from the truncation errors given in Sects. 3.2 and 3.3, that more accurate solutions could be obtained on a refined grid. These aspects are considered further in Sect. 4.4. However for a given required solution accuracy it may be more economical to solve a higher-order finite difference scheme on a coarse grid than a low-order scheme on a finer grid. This leads to the concept of computational efficiency which is examined in Sect. 4.5. An important question concerning computational solutions is what guarantee can be given that the computational solution will be close to the exact solution of the partial differential equation(s) and under what circumstances the computational solution will coincide with the exact solution. The second part of this question can be answered (superficially) by requiring that the approximate (computational) solution should converge to the exact solution as the grid spacings At, Ax shrink to zero (Sect. 4.1). However, convergence is very difficult to establish directly so that an indirect route, as indicated in Fig. 4.1, is usually followed. The indirect route requires that the system of algebraic equations formed by the discretisation process (Sect. 3.1) should be consistent (Sect. 4.2) with the governing partial differential equation(s). Consistency implies that the discretisation process can be reversed, through a Taylor series expansion, to recover the governing equation(s). In addition, the algorithm used to solve the algebraic equations to give the approximate solution, T, must be stable (Sect. 4.3). Then the pseudo-equation. is invoked to imply convergence. The conditions under which (4.1) can be made precise are given by the Lax equivalence theorem (Sect. 4.1.1). It is very difficult to obtain theoretical guidance for the behaviour of the solution on a grid of finite size. Most of the useful theoretical results are strictly only applicable in the limit that the grid size shrinks to zero. However the connections that are established between convergence (Sect. 4.1), consistency (Sect. 4.2) and stability (Sect. 4.3) are also qualitatively useful in assessing computational solutions on a finite grid.
For the equations that govern fluid flow, convergence is usually impossible to demonstrate theoretically. However, for problems that possess an exact solution, like the diffusion equation, it is possible to obtain numerical solutions on a successively refined grid and compute a solution error. Convergence implies that the solution error should reduce to zero as the grid spacing is shrunk to zero. For program DIFF (Fig. 3.13), solutions have been obtained on successively refined spatial grids, Ax =0.2, 0.1, 0.05 and 0.025. The corresponding rms errors are shown in Table 4.1 for s=0.50 and 0.30. It is clear that the rms error reduces like Ax2 approximately. Based on these results it would be a reasonable inference that refining the grid would produce a further reduction in the rms error and, in the limit of Ax (for fixed s) going to zero, the solution of the algebraic equations would converge to the exact solution. The establishment of numerical convergence is rather an expensive process since usually very fine grids are necessary. As s is kept constant in the above example the timestep is being reduced by a factor of four for each halving of Ax. In Table 4.1 the solution error is computed at t = 5000 s. This implies the finest grid solution at s = 0.30 requires 266 time steps before the solution error is computed. For the diffusion equation (3.1) with zero boundary values and initial value T (x, 0) = sin(nx), 0 ~ x ~ 1, the rms solution error lelrms is plotted against grid . spacing A x in Fig. 4.2. The increased rate of convergence (fourth-order convergence) for s =i, compared with other values of s~! (second-order convergence), is clearly seen, i.e. the convergence rate is like AX4 for s=i, and like AX2 otherwise. As will be demonstrated in Sect. 4.2, the superior convergence rate for s=i is to be expected from a consideration of the leading term in the truncation error. Typically, for sufficiently small grid spacings A x, A t, the solution error will reduce like the truncation error as deltax, deltat _>0.
This is the tendency for any spontaneous perturbations (such as round-off error) in the solution of the system of algebraic equations (Figs. 3.1 and 4.1) to decay. A stable solution produced by the FTCS scheme with s = 0.5 is shown in Fig. 3.15. A typical unstable result (s = 0.6) is shown in Fig. 4.3. These results have been obtained with Ll x = 0.1 and the same initial and boundary conditions as used to generate Fig. 3.15. It is clear from Fig. 4.3 that an unphysical oscillation originates on the line of symmetry and propagates to the boundaries. The amplitude of the oscillation grows with increasing time. The concept of stability is concerned with the growth, or decay, of errors introduced at any stage of the computation. In this context, the errors referred to are not those produced by incorrect logic but those which occur because the computer cannot give answers to an infinite number of decimal places. In practice, each calculation made on the computer is carried out to a finite number of significant figures which introduces a round-off error at every step of the computation. Hence the computational solution to (3.41) is not T't 1, but * r;+ 1, the numerical solution of the system of algebraic equations. A particular method is stable if the cumulative effect of all the round-off errors produced in the application of the algorithm is negligible. More specifically, consider the errors ej= Tj-*Tj (4.15) introduced at grid points (j, n), where j = 2, 3, ... , J -1 and n = 0, 1, 2. It is usually not possible to determine the exact value of the numerical error ej at the (j, n)-th grid point for an arbitrary distribution of errors at other grid points. However, it can be estimated using certain standard methods, some of which will be discussed in this section. In practice, the numerical solutions are typically more accurate than these estimates indicate, because stability analyses often assume the worst possible combination of individual errors. For instance, it may be assumed that all errors have a distribution of signs so that their total effect is additive, which is not always the case. It can be shown that, for linear algebraic equations produced by discretisation, the corresponding error terms satisfy the same homogeneous algebraic equations as the values of T. For instance, using the FTCS scheme (3.41) means that we are actually calculating *T~+ 1 using *T~ *T'! and *T'! so that J J-l' J J+l, (4.16) Substitution of (4.15) into (4.16), followed by application of (3.41), which applies since the exact solutions ofthe algebraic equations, Tj, satisfy the FTCS algorithm, yields the homogeneous algebraic equation (4.17) Assuming given boundary and initial values, the initial errors, eJ, j = 2, 3, ... , J -1, and the boundary errors e7 and ej, n = 0, 1, 2, ... for this equation, will all be zero. Unless some (round-off) error is introduced in calculating the value of Tj at some interior node, the resulting errors in the solution will remain zero. The two most common methods of stability analysis are the matrix method and the von Neumann method. Both methods are based on predicting whether there will be a growth in the error between the true solution of the numerical algorithm and the actually computed solution, i.e. including round-off contamination. An alternative interpretation of stability analysis is to suppose that the initial conditions are represented by a Fourier series. Each harmonic or mode in the Fourier series will grow or decay depending on the discretised equation, which typically furnishes a specific expression for the growth (or decay) factor for each mode. If a particular mode can grow without bound, the discretised equation has an unstable solution. This interpretation of stability (Richtmyer and Morton 1967, pp.9-13) is exploited directly in the von Neumann method of stability analysis (Sects. 4.3.4 and 4.3.5). The unbounded growth of a particular mode is still possible if the discretised equations are solved exactly, i.e. with no (round-off) errors being present. If (round-off) errors are introduced the same unstable nature of the discretised equations will cause unacceptable growth of the errors. Consequently the procedures for analysing the stability of the discretised equations are the same irrespective of the manifestation of the inherent stability or instability.
It is clear that as Llt tends to zero, Ej tends to zero and (4.13) coincides with the governing equation. Consequently (4.8) is consistent with the governing equation. In (4.14) all spatial derivatives have been converted to equivalent tIme derivatives. Using (4.12) it would be possible to express the truncation error in terms of the spatial grid size and derivatives only, as in (4.7). A comparison of (4.14) and (4.7) indicates that there is no choice of s that will reduce the truncation error of the fully implicit scheme to O(LlX4). It might appear from the above two examples that consistency can be taken for granted. However, attempts to construct algorithms that are both accurate and stable can sometimes generate potentially inconsistent discretisations, e.g. the DuFort-Frankel scheme, Sect. 7.1.2.
For explicit methods a single unknown, e.g. Tj + 1 , appears on the left hand side of the algebraic formula resulting from discretisation.
Effectively this scheme evaluates the spatial derivative at the average of the nth and (n + l}-th time levels, i.e. at the (n + 1/2)-th time level. If a Taylor expansion is made about (j, n + 1/2), (7.22) is found to be consistent with (7.1) with a truncation error of O(Llt2 , Llx2 ). This is a considerable improvement over the fully implicit and FTCS schemes that are only first-order accurate in time. A von Neumann stability analysis indicates that the Crank-Nicolson scheme is unconditionally stable, Table 7.1. A rearrangement of (7.22) gives the algorithm -0.5s Tj:!:t +(1 +s) Tj+1-0.5s Tj:t =0.5s Tj-1 +(I-s) Tj+0.5s Tj+1 , (7.23) which may be compared with (7.20). By considering all spatial nodes (7.23) produces a tridiagonal system of equations which can be solved efficiently using the Thomas algorithm Because ofthe second-order temporal accuracy, the Crank-Nicolson scheme is a very popular method for solving parabolic partial differential equations. The properties of the Crank-Nicolson scheme are summarised in Table 7.1. A generalisation of (7.22) can be obtained by writing (7.24) where A Tj + 1 = Tr 1 - Tj and 0 ~ 13 ~ 1. If 13 = 0 the FTCS scheme is obtained. If 13=0.5 the Crank-Nicolson scheme is obtained and if 13= 1.0 the fully implicit scheme is obtained. A von Neumann stability analysis of (7.24) indicates that a stable solution is possible for At ~ (X(I- 213) if 0 ~ 13 < 1/2 no restriction if 1/2~f3~ 1 . It may be noted that the Crank-Nicolson scheme is on the boundary of the unconditionally stable regime. For many steady flow problems it is efficient to solve an equivalent transient problem until the solution no longer changes (Sect. 6.4). However, often the solution in different parts of the computational domain approaches the steady-state solution at significantly different rates; the equations are then said to be stiff (Sect. 7.4). Unfortunately the Crank-Nicolson scheme often produces an oscillatory solution in this situation which, although stable, does not approach the steady state rapidly. Certain three-level (in time) schemes are more effective than the Crank-Nicolson scheme in this regard.
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