Question

(a) Show that the application of an explicit \(s\) -stage RK method to the test equation can be written as \(y_{i+1}=R(z) y_{i}\), where \(z=\lambda h\) and \(R(z)\) is a polynomial of degree at most \(s\) in \(z\) (b) Further, if the order of the method is \(q=s\), then any such method for a fixed \(s\) has the same amplification factor \(R(z)\), given by $$ R(z)=\sum_{j=0}^{s} \frac{z^{j}}{j !} $$ [Hint: Consider the Taylor expansion of the exact solution of the test equation.]

Short answer

Question: Apply an explicit s-stage RK method to the test equation \(y' = \lambda y\), and show that if the order of the method is q=s, then any such method for a fixed s has the same amplification factor \(R(z)\), given by \(R(z) = \sum_{j=0}^{s} \frac{z^{j}}{j!}\). Answer: Applying an explicit s-stage RK method to the test equation \(y' = \lambda y\) results in the following amplification factor: \(R(z) = 1 + \sum_{i=1}^s b_i z + \sum_{i=1}^s \sum_{j=1}^s b_i a_{ij} z^2\). If the order of an s-stage RK method is q=s, any such method for a fixed s has the same amplification factor \(R(z) = \sum_{j=0}^{s} \frac{z^{j}}{j!}\).

Step-by-step solution

(a) Apply explicit s-stage RK method to test equation and find the amplification factor

Consider the test equation \(y' = \lambda y\) with a step size of h. Let's apply an s-stage RK method to this test equation. An s-stage RK method can be defined by coefficients \(a_{ij}, b_i,\) and \(c_i\) for i, j = 1, 2, ..., s. The method can be written as: $$ \begin{aligned} k_i &= \lambda (y_n + h \sum_{j=1}^s a_{ij} k_j), \quad i=1,2,\dots, s \\ y_{n+1} &= y_n + h \sum_{i=1}^s b_i k_i. \end{aligned} $$ Now, we can substitute the definition of \(k_i\) back into the equation for \(y_{n+1}\) and simplify: $$ \begin{aligned} y_{n+1} &= y_n + h \sum_{i=1}^s b_i \lambda (y_n + h \sum_{j=1}^s a_{ij} k_j) \\ &= y_n (1 + \lambda h \sum_{i=1}^s b_i) + h^2 \lambda \sum_{i=1}^s \sum_{j=1}^s b_i a_{ij} k_j \\ &= y_n R(z), \end{aligned} $$ where \(z = \lambda h\) and \(R(z)\) is a polynomial of degree at most s in z given by \(R(z) = 1 + \sum_{i=1}^s b_i z + \sum_{i=1}^s \sum_{j=1}^s b_i a_{ij} z^2\).

(b) Show that for q=s, all methods have the same amplification factor.

Since the exact solution of the test equation can be written as \(y(t) = y(0)e^{\lambda(t)}\), its Taylor expansion is given by: $$ y(t) = y(0) (1 + \lambda t + \frac{(\lambda t)^2}{2!} + \frac{(\lambda t)^3}{3!} + \cdots + \frac{(\lambda t)^s}{s!}). $$ For an s-stage RK method with order q = s, we expect the error to be given by the first s terms of the Taylor expansion for the exact solution. Considering the Taylor expansion of the exact solution, for an s-step RK method, after s steps we have: $$ y(t + sh) = y(0) (1 + \lambda sh + \frac{(\lambda sh)^2}{2!} + \cdots + \frac{(\lambda sh)^s}{s!}). $$ Replace \(z\) with \(\lambda h\) and compare with the form of \(R(z)\) as found in part (a): $$ R(z) = \sum_{j=0}^s y_0 \frac{z^j}{j!}. $$ Thus, we have shown that if the order of an s-stage RK method is q = s, then any such method for a fixed s has the same amplification factor \(R(z)\), given by \(R(z) = \sum_{j=0}^{s} \frac{z^{j}}{j!}\).

Key concepts

Amplification Factor

The amplification factor, often denoted as \(R(z)\), plays a critical role in analyzing the behavior of numerical methods, especially when applied to differential equations. In the context of Runge-Kutta methods and the given problem, the amplification factor dictates how the solution evolves from one time step to the next. Runge-Kutta methods are widely used for their accuracy in solving ordinary differential equations. In essence, the amplification factor determines the scaling of the solution between successive steps.

For a method to be effective, the amplification factor should accurately approximate the behavior of the exponential growth \(e^{\lambda t}\) of the test equation over each time step. This is achieved by comparing \(R(z)\) to the Taylor Expansion of the exponential function, aiming for them to match closely over the desired range of steps. The closer \(R(z)\) approximates the exponential function, the more accurate the solution will be.

• Amplification factor \(R(z)\) is typically a polynomial in \(z\).
• It is crucial for determining numerical stability and accuracy.
• In the problem, \(R(z)\) for an \(s\)-stage RK method becomes \(R(z)=\sum_{j=0}^{s} \frac{z^{j}}{j!}\), which is a polynomial form resembling the Taylor series of \(e^{z}\).

Polynomial Order

Polynomial order in numerical analysis generally refers to the degree of the polynomial used in the numerical method. In the context of the Runge-Kutta (RK) methods discussed here, polynomial order has significant implications for both the accuracy and the complexity of the computation involved.

The order of the polynomial \(R(z)\) is directly tied to the number of stages \(s\) used in the RK method. As explained in the problem, for an \(s\)-stage Runge-Kutta method, the resulting polynomial in \(z\) is of at most degree \(s\). This degree represents the highest power of \(z\) and therefore the highest order of local truncation error that the method can accurately capture.

• Higher polynomial orders typically suggest a more accurate approximation over smaller time steps.
• For \(s\)-stage methods, \(R(z)\) becomes a polynomial of degree at most \(s\), allowing the method to represent derivatives up to the \(s^{th}\) degree.
• Higher order polynomials tend to better approximate the Taylor expansion of the exponential function, improving results for more complex or larger problems.

Taylor Expansion

The Taylor Expansion is a fundamental concept in calculus, providing a powerful method for approximating complex functions. It expresses a function as an infinite sum of terms calculated from derivatives at a single point. When dealing with differential equations, the Taylor Expansion helps to approximate solutions over short intervals by incrementally adding more terms for increased precision.

In the problem context, the exact solution of the test equation \(y'(t) = \lambda y(t)\), an exponential function, is approximated by using its Taylor Expansion. The expansion takes the form:
\[y(t) = y(0)\left(1 + \lambda t + \frac{(\lambda t)^2}{2!} + \frac{(\lambda t)^3}{3!} + \cdots\right)\]

By comparing the polynomial form of the amplification factor \(R(z)\) with the Taylor expansion, we can see how well the Runge-Kutta method mimics the true behavior of the system it models. The accuracy of this correspondence underlines the method's validity. Higher order terms in the expansion are considered to better approximate solution behavior among numerical methods.

• Provides a sequence of approximations that converge to the exact function.
• For RK methods, the Taylor Expansion allows us to derive accurate expressions for \(R(z)\).
• Crucial for demonstrating that RK methods, especially when \(q=s\), yield identical amplification factors.

Numerical Stability

Numerical stability is a crucial consideration when applying iterative numerical methods, like Runge-Kutta, to solve differential equations. A numerically stable method ensures that errors do not multiply uncontrollably with each step of the algorithm.

In terms of Runge-Kutta methods, numerical stability can often be assessed by examining the amplification factor \(R(z)\). Stability requires that the absolute value of the amplification factor remains less than or equal to one for certain ranges of \(z\), meaning the growth of errors is controlled.

• Numerically stable methods yield reliable solutions over large integration periods.
• Control of \(|R(z)|\) indicates proper handling of errors.
• Stability regions are analyzed to ensure method suitability for given problems.

When choosing parameters for numerical simulations, ensuring numerical stability can mean the difference between divergent solutions and those that converge on a correct, meaningful answer. Therefore, understanding and ensuring numerical stability is critical for the successful application of Runge-Kutta methods to real-world problems.