Problem 11 - A First Course in the Numerical Analysis of Differential Equations

Question

Apply the Fourier stability technique to the FD scheme $$ \begin{aligned} u_{\ell}^{n+1}=& \frac{1}{2}\left(2-5 \mu+6 \mu^{2}\right) u_{\ell}^{n}+\frac{2}{3} \mu(2-3 \mu)\left(u_{\ell-1}^{n}+u_{\ell+1}^{n}\right) \\ &-\frac{1}{12} \mu(1-6 \mu)\left(u_{\ell-2}^{n}+u_{\ell+2}^{n}\right), \quad \ell \in \mathbb{Z} \end{aligned} $$ You should find that stability occurs if and only if \(0 \leq \mu \leq \frac{2}{3}\). [We have not specified which equation - if any - the scheme is supposed to solve, but this, of course, has no bearing on the question of stability.].

Step-by-Step Solution

Answer

Stability occurs if and only if \( 0 \leq \mu \leq \frac{2}{3} \).

Step 1: Assume a Solution Form

To apply Fourier stability analysis, assume the solution takes the form of a Fourier mode: \[ u_{\ell}^{n} = \xi^{n} e^{i k \ell} \]where \( \xi \) is the amplification factor, \( i \) is the imaginary unit, \( k \) is the wave number, and \( \ell \) is the spatial index.

Step 2: Substitute into the Scheme

Substitute the assumed solution form into the finite difference scheme:\[u^{n+1} = \frac{1}{2}(2-5\mu+6\mu^2)u^{n} + \frac{2}{3}\mu(2-3\mu)(u^{n}_{\ell-1} + u^{n}_{\ell+1}) -\frac{1}{12}\mu(1-6\mu)(u^{n}_{\ell-2} + u^{n}_{\ell+2})\]Plug in \( u_{\ell}^{n} = \xi^{n} e^{i k \ell} \), resulting in expressions involving \( \xi \) and exponential terms.

Step 3: Compute Amplification Factor

Compute the amplification factor \( \xi \) from the substituted expression:\[ \xi = \frac{1}{2}(2-5\mu+6\mu^2) + \frac{2}{3}\mu(2-3\mu)(e^{-ik} + e^{ik}) - \frac{1}{12}\mu(1-6\mu)(e^{-2ik} + e^{2ik}) \].Utilize the identities \( e^{ik} + e^{-ik} = 2\cos(k) \) and \( e^{2ik} + e^{-2ik} = 2\cos(2k) \) to simplify.

Step 4: Simplify with Trigonometric Identities

Incorporate trigonometric identities:\[ \xi = \frac{1}{2}(2-5\mu+6\mu^2) + \frac{4}{3}\mu(2-3\mu)\cos(k) - \frac{1}{6}\mu(1-6\mu)\cos(2k) \].This provides a compact form for further analysis.

Step 5: Set Stability Condition

The scheme is stable if and only if the magnitude of \( \xi \) is less than or equal to 1 for all wave numbers \( k \):\[ |\xi| \leq 1 \].Examine this inequality to find bounds on \( \mu \).

Step 6: Analyze Extremes of \( \xi \)

Focus on extreme cases, e.g., \( k = 0 \) and \( |\cos(k)| = 1 \), as these scenarios often provide critical conditions for stability.Consider the cases for minimal and maximal \( |\xi| \) and solve for \( \mu \).

Step 7: Determine Stability Range for \( \mu \)

Upon solving, impose that the maxima of \( |\xi| \) does not exceed 1. Relax boundaries and solve inequalities determined in Step 6 to conclude that stability range is:\[ 0 \leq \mu \leq \frac{2}{3} \].

Key Concepts

Finite Difference Scheme

The finite difference scheme is a numerical method used to approximate solutions to differential equations. This approach involves replacing derivatives with difference equations, turning an otherwise continuous problem into one that can be solved with algebra. A scheme's formation typically depends on the spatial and temporal points of the problem, creating a grid. Each grid point updates its value according to the scheme's provided formula.

For the specific problem given, the scheme operates on the basis of a mathematical expression incorporating coefficients that depend on a parameter, \( \mu \). This particular scheme is designed to evolve the solution from time level \( n \) to \( n+1 \), using the values from neighboring spatial points. Understanding how to apply these schemes is essential for accurately simulating the behavior of physical systems.

Stability Condition

In numerical analysis, the stability condition determines whether errors in calculations dissipate over time or grow uncontrollably. A numerical scheme is said to be stable if the error doesn't amplify as iterations progress. For Fourier analysis, stability is assessed by ensuring that the amplification factor remains within specific bounds.

The general rule of thumb is to analyze the amplification factor \( \xi \) to see if its magnitude is less than or equal to one. This ensures that as we iterate the calculations over time, deviations in results don't lead to non-physical predictions. In the given exercise, detailed investigation shows that the scheme is stable when the parameter \( \mu \) is constrained between \( 0 \) and \( \frac{2}{3} \). This is because, for these values, the maximum value of \( |\xi| \) remains at most 1, thus preserving the stability of the scheme.

Amplification Factor

The amplification factor, symbolized as \( \xi \), indicates how much the solution is amplified or dampened at each time step. It is a crucial element in understanding the stability of a finite difference scheme. In the context of Fourier stability analysis, \( \xi \) is derived from substituting the assumed form of the solution into the finite difference scheme.

In the exercise, the form of \( \xi \) is a function that involves both trigonometric and algebraic components. Calculating \( \xi \) involves substituting a trial solution into the scheme and simplifying using trigonometric identities. The magnitude of \( \xi \) and its behavior over successive time steps are key factors in drawing stability conclusions. In particular, if \( |\xi| \leq 1 \), the scheme does not let errors grow, ensuring a stable solution.

Trigonometric Identities

Trigonometric identities play a significant role in simplifying expressions within the finite difference scheme. They are crucial in transforming the derived expressions from exponential form into a more manageable trigonometric form. Two common identities used in this context are:

• \( e^{ik} + e^{-ik} = 2\cos(k) \)
• \( e^{2ik} + e^{-2ik} = 2\cos(2k) \)

These identities are vital because they allow the conversion of complex exponential expressions into simple cosine terms, making the amplification factor easier to analyze. In this exercise, after substituting the assumed Fourier mode solution into the scheme, these identities help in expressing the amplification factor in terms of cosines, which directly ties into investigating the stability condition.