Question

Continuing with the notation of Exercise 15, consider approximating \(q\left(x_{i}\right)\) in terms of values of \(c(x)\) and \(g(x)\), where $$ q(x)=-\left[c(x) g^{\prime}(x)\right]^{\prime} $$ is known to be square integrable (but not necessarily differentiable) on \([0, \pi]\). The function \(g\) is assumed given, and it has some jumps of its own to offset those of \(c(x)\) so as to create a smoother function \(\phi(x)=c(x) g^{\prime}(x)\). The latter is often termed the flux function in applications. (a) Convince yourself yet again that Chebyshev and Fourier differentiations on the entire interval \([0, \pi]\) are not the way to go. (b) Evaluate the merits (or lack thereof) of the difference approximation $$ h^{-1}\left[\frac{c_{i+1 / 2}\left(g_{i+1}-g_{i}\right)}{h}-\frac{c_{i-1 / 2}\left(g_{i}-g_{i-1}\right)}{h}\right] $$ with \(g_{i}, g_{i \pm 1}\) and \(c_{i \pm 1 / 2}\) appropriately defined for \(i=1, \ldots, n-1\). (c) We could instead write \(q(x)=-c(x) g^{\prime \prime}(x)-c^{\prime}(x) g^{\prime}(x)\) and discretize the latter expression. Is this a good idea? Explain.

Short answer

Answer: Chebyshev and Fourier differentiations might not be suitable for approximating the function q(x) in this exercise because q(x) is not necessarily differentiable and could have discontinuities on the interval [0, π]. These differentiation methods work best with smooth and periodic functions, which is not guaranteed in this case. As a result, using these methods could lead to erroneous approximations.

Step-by-step solution

(a) Inapplicability of Chebyshev and Fourier differentiations

Chebyshev and Fourier differentiations may not be suitable in this case because \(q(x)\) is not necessarily differentiable and could have discontinuities on the interval \([0, \pi]\). These differentiation methods work best with smooth and periodic functions, which is not guaranteed in this case. As a result, using these methods could lead to erroneous approximations.

(b) Evaluating the given difference approximation

The given difference approximation is: $$ h^{-1}\left[\frac{c_{i+1 / 2}\left(g_{i+1}-g_{i}\right)}{h}-\frac{c_{i-1 / 2}\left(g_{i}-g_{i-1}\right)}{h}\right]. $$ This approximation seems to be a better suited method for \(q(x)\) compared to Chebyshev and Fourier differentiations since it captures the possibly discontinuous behavior. The approximation uses midpoint values of \(c\) along with differences of the function \(g\) at adjacent points and hence, provides a discrete estimate of the flux function.

(c) Discretizing an alternative expression of \(q(x)\)

Consider the alternative expression for \(q(x)\): $$ q(x)=-c(x) g^{\prime \prime}(x)-c^{\prime}(x) g^{\prime}(x). $$ Discretizing this expression may require finding the second derivative of \(g(x)\) and the first derivative of \(c(x)\). Since \(q(x)\) is not necessarily differentiable, this could introduce inaccuracies in the approximation. Moreover, accurately approximating the derivatives of these functions could become computationally expensive, especially if the functions have many discontinuities. Therefore, discretizing the alternative expression of \(q(x)\) might not be a good idea compared to the given difference approximation.

Key concepts

Square Integrable Functions

Square integrable functions, or L2 functions, are an essential concept in mathematical analysis and physics.
A function f is considered square integrable on an interval, say [a, b], if the integral of the square of the absolute value of f over that interval is finite, as expressed by the following equation:\[\begin{equation}\int_a^b |f(x)|^2 dx < +\infty\end{equation}\]In simpler terms, you can think of these functions as having a finite 'energy' over the interval. This property is crucial when dealing with physical applications where infinite energy is not realistic.
For the problem at hand, knowing that the function q(x) is square integrable on [0, π] tells us that despite its potential discontinuities, it is mathematically 'well-behaved' enough to assure the existence of certain types of solutions and simplifies the analysis of the system.

Flux Function

The term flux function, \[\begin{equation}\phi(x) = c(x)g’(x),\end{equation}\]in this context, represents a physical quantity that deals with the flow or transfer of a property, such as energy or matter, through a surface or along a curve.
For functions c(x) and g(x), the flux function helps to model the intensity and direction of this flow, which could correspond to phenomena like heat transfer or fluid dynamics. As stated in the exercise, the flux function \[\begin{equation}\phi(x)\end{equation}\]is assumed to be smoother than the individual functions c(x) and g(x), due to the contrasting behaviors compensating each other. This tends to be important in numerical methods where smoother functions often lead to more stable and accurate approximations.

Difference Approximation

Difference approximation is a numerical technique used to estimate derivatives of functions.
This process involves computing the differences between function values at several discrete points. The exercise introduces an approximation for the function q(x):\[\begin{equation}h^{-1}\left[\frac{c_{i+1 / 2}(g_{i+1}-g_{i})}{h}-\frac{c_{i-1 / 2}(g_{i}-g_{i-1})}{h}\right]\end{equation}\]Where h is a small step size and the subscript i denotes discrete evaluation points.
This type of approximation is particularly useful when the function being differentiated is not smooth since traditional differentiation requires smoothness. In our case, the difference approximation aligns well with the physical intuition of the flux function and is beneficial when handling functions that may have jumps or be piecewise continuous.

Chebyshev Differentiation

Chebyshev differentiation is a spectral method of estimating derivatives using Chebyshev polynomials.
These polynomials form an orthogonal basis for the space of functions defined on the interval [-1, 1]. The method is highly accurate for smooth functions that are well-approximated by polynomials.However, Chebyshev differentiation may not be suitable for the function q(x) given in the exercise because it is not guaranteed to be differentiable or smooth across the domain [0, π]. Using Chebyshev polynomials for such functions can lead to large errors, rendering the method ineffective for our specific application. This emphasizes the importance of aligning the numerical method with the function's characteristics to ensure accurate and reliable results. In cases like this, where discontinuities are present, piecewise or non-polynomial approximations like the difference method may be more reliable.

Fourier Differentiation

Fourier differentiation is another spectral method where the Fourier series is used to represent a function and estimate its derivative. Like Chebyshev differentiation, it is most effective for periodic and smooth functions.The key to Fourier differentiation is that by expressing a function in terms of sinusoidal waves, we can easily compute derivatives by multiplying by factors dependent on the wave's frequency. But when a function exhibits discontinuities, as may be the case with q(x), the Fourier series may not converge well, introducing artifacts known as Gibbs oscillations.Therefore, caution is advised when applying Fourier differentiation to non-periodic or discontinuous functions, further accentuating the choice of using the difference approximation in our problem scenario for a more suitable and accurate approach to modeling physical phenomena described by square integrable functions.