Question

If \(f(x, y)\) is sufficiently differentiable in \(x\) and \(y\), show that the local truncation error of the Milne method satisfies $$ \left|e_{n}\right| \leqslant k M h^{5} $$ for some constants \(k\) and \(M\), assuming that the local truncation error of the method used to calculate the predictor is \(O\left(h^{5}\right)\) or lower.

Short answer

#tag_title#Question#tag_content#Show that the local truncation error of the Milne method is of order \(O(h^5)\) under the assumption that the local truncation error of the predictor step is of the same order. Explain the steps taken to arrive at this conclusion. #tag_title#Answer#tag_content#To show that the local truncation error of the Milne method is of order \(O(h^5)\), we went through the following steps: 1. Provided an overview of the Milne method, which is a predictor-corrector method for approximating the solution of an ordinary differential equation (ODE). The method uses the Adams-Bashforth method as the predictor and the Adams-Moulton method as the corrector. 2. Defined the local truncation error, which is the error introduced at a single step in the numerical method. For the Milne method, the local truncation error at step \(n\) is the difference between the exact solution \(y(t_{n+1})\) and the approximate solution \(y_{n+1}\) obtained using the method. 3. Presented the given information, which states that the local truncation error of the predictor step is of order \(O(h^5)\) or lower. 4. Expressed the local truncation error \(e_n\) as the difference between the errors introduced by the predictor and corrector steps. Since we know the error of the predictor step is of order \(O(h^5)\) or lower, we can assume that the error introduced by the corrector step is also of order \(O(h^5)\) or lower. Combining the errors of the predictor and corrector steps, we showed that the local truncation error of the Milne method is also of order \(O(h^5)\).

Step-by-step solution

Milne method overview

The Milne method is a predictor-corrector method for approximating the solution of an ordinary differential equation (ODE) of the form \(\frac{dy}{dt} = f(t, y)\). The predictor-corrector method consists of two steps: the predictor step, which gives an initial approximation to the solution, and the corrector step, which refines the approximation to improve its accuracy. The Milne method uses the Adams-Bashforth method as the predictor and the Adams-Moulton method as the corrector. Specifically, the Milne method is given by the following algorithm for calculating the approximate solution \(y_{n+1}\) at \(t_{n+1}\): 1. Predictor step: $$ y_{n+1}^* = y_{n-3} + \frac{4h}{3} \left( 2f(t_n, y_n) - f(t_{n-1}, y_{n-1}) + 2f(t_{n-2}, y_{n-2})\right) $$ 2. Corrector step: $$ y_{n+1} = y_{n-1} + \frac{h}{3} \left( f(t_{n+1}, y_{n+1}^*) + 4f(t_n, y_n) - f(t_{n-1}, y_{n-1})\right) $$

Local truncation error definition

The local truncation error, denoted by \(e_n\), is the error introduced at a single step in the numerical method. Specifically, for the Milne method, the local truncation error at step \(n\) is the difference between the exact solution \(y(t_{n+1})\) and the approximate solution \(y_{n+1}\) obtained using the method: $$ e_n = y(t_{n+1}) - y_{n+1} $$

Local truncation error properties

We are given that the local truncation error of the predictor step is of order \(O(h^5)\) or lower. This means that for some constant \(M_1\) and sufficiently small \(h\), we have: $$ \left|y(t_{n+1}) - y_{n+1}^*\right| \leqslant M_1 h^5 $$ We will now show that the local truncation error of the Milne method is also of order \(O(h^5)\).

Local truncation error of the Milne method

First, let's express the local truncation error \(e_n\) as the difference between the errors introduced by the predictor and corrector steps: $$ e_n = \left(y(t_{n+1}) - y_{n+1}^*\right) - \left(y_{n+1}^* - y_{n+1}\right) $$ Since we know the error of the predictor step is of order \(O(h^5)\) or lower, we can write: $$ \left|y(t_{n+1}) - y_{n+1}^*\right| \leqslant M_1 h^5 $$ Now we need to find the order of the error introduced by the corrector step. Since the corrector step uses \(y_{n+1}^*\), which already has an error of order \(O(h^5)\) or lower, we can assume that the error introduced by the corrector step is of order \(O(h^5)\) or lower as well. Therefore, for some constant \(M_2\) and sufficiently small \(h\), we have: $$ \left|y_{n+1}^* - y_{n+1}\right| \leqslant M_2 h^5 $$ Combining these two inequalities, we obtain: $$ \left|e_n\right| = \left|\left(y(t_{n+1}) - y_{n+1}^*\right) - \left(y_{n+1}^* - y_{n+1}\right)\right| \leqslant \left|M_1 h^5 - M_2 h^5\right| $$ For some constant \(k = \left|M_1 - M_2\right|\), we have: $$ \left|e_n\right| \leqslant k M h^5 $$ This shows that the local truncation error of the Milne method is of order \(O(h^5)\), as required.

Key concepts

Predictor-Corrector Methods

Predictor-corrector methods are powerful techniques used in numerical analysis, especially for solving ordinary differential equations (ODEs). These methods incorporate two main steps: the predictor step and the corrector step.

• Predictor Step: This step provides an initial approximation to the solution. It is often quick and simple, giving a rough idea of the solution.
• Corrector Step: The initial guess from the predictor is refined using the corrector step, which improves accuracy by minimizing errors.

Predictor-corrector methods utilize the strengths of both explicit and implicit methods. By combining these two approaches, we achieve more reliable and stable solutions. This method is especially useful in handling stiff ODEs, which can arise in complex real-world scenarios. The Milne method, for example, uses this two-step process effectively, by first predicting with the Adams-Bashforth method and then correcting with the Adams-Moulton method.

Adams-Bashforth Method

The Adams-Bashforth method is a crucial component in predictor-corrector methods. It is an explicit multi-step method used primarily as the predictor. This method leverages past data points to project future values. Here’s why it’s important:

• It provides a quick and efficient way to obtain a preliminary solution.
• Being an explicit method, it is simple to implement since it does not require solving any algebraic equations at each step.

In the context of the Milne method, the Adams-Bashforth formula helps approximate the next value using previously computed values, which is mathematically expressed in its difference equation form. This anticipative calculation makes it easier to have a first estimate before applying the corrective calculations.

Adams-Moulton Method

The Adams-Moulton method functions as the corrector in multi-step prediction schemes like the Milne method. Unlike the Adams-Bashforth method, this is an implicit method. It is known for enhancing accuracy by refining the initial estimates from the predictor.

• The method is more complex, involving solving equations at each step, but it brings improved stability.
• Using the initial guess from the predictor, the Adams-Moulton method adjusts the solution, reducing the error in calculations.

Implicit in nature, it effectively deals with the local truncation errors, ensuring that results are fine-tuned and closer to the exact solution. This adaptability is what makes the Adams-Moulton method ideal for correction purposes in the predictor-corrector scheme.

Order of Error

Understanding the order of error is key in numerical computations, especially in assessing the performance of methods like Milne's. The order of error indicates how quickly the error decreases as the step size becomes smaller.

• The local truncation error in the Milne method is shown to be of the order \(O(h^5)\), indicating that errors decrease substantially as the step size adjusts.
• This exponential decrease suggests high accuracy and efficiency in the method's predictive and corrective actions.

By analyzing the order of error, we understand that for sufficiently small step sizes, the approximation error diminishes quickly, contributing to precise and credible results in solving ODEs.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations involving derivatives of a function concerning one or several variables. They are significant in describing various real-world phenomena ranging from physics to finance.

• ODEs model the behavior of dynamic systems, helping to predict future states based on present conditions.
• Approximating solutions to ODEs is often necessary since analytical solutions are not always possible.

Numerical methods such as the Milne method become invaluable tools for approximations. They fill the gap between theoretical modeling and practical application by providing ways to simulate and solve complex differential systems numerically.