Question

(a) Use the Taylor expansion $$ \begin{aligned} y\left(t_{i+1}\right)=& y\left(t_{i}\right)+h y^{\prime}\left(t_{i}\right)+\frac{h^{2}}{2} y^{\prime \prime}\left(t_{i}\right)+\frac{h^{3}}{6} y^{\prime \prime \prime}\left(t_{i}\right) \\ &+\frac{h^{4}}{24} y^{(i v)}\left(t_{i}\right)+\frac{h^{5}}{120} y^{(v)}\left(t_{i}\right)+\mathcal{O}\left(h^{6}\right) \end{aligned} $$ to derive a corresponding series expansion for the local truncation error of the forward Euler method. (b) Manipulating the forward Euler method written for the step sizes \(h\) and \(h / 2\), apply extrapolation (Section 14.2) to obtain a second order one-step method. (c) Manipulating the forward Euler method written for the step sizes \(h, h / 2\), and \(h / 3\), apply extrapolation to obtain a third order one-step method.

Short answer

As this is a short answer question, we can write the final solutions as follows: (a) The local truncation error for the forward Euler method is given by: $$ E(t_i) = \frac{h^2}{2}y''(t_i) + \frac{h^3}{6}y'''(t_i) + \frac{h^4}{24}y^{(4)}(t_i) + \frac{h^5}{120}y^{(5)}(t_i) + \mathcal{O}(h^6) $$ (b) The second-order one-step method using extrapolation is given by: $$ y(t_{i+1}) = \frac{1}{2}\left[2y^{(h/2)}(t_i) - y^{(h)}(t_i) + E^{(h)}(t_i) - E^{(h/2)}(t_i)\right] $$ (c) The third-order one-step method using extrapolation is given by: $$ y(t_{i+1}) = \frac{1}{8}\left[\left(2y^{(h/2)}(t_i) + 4y^{(h/3)}(t_i) - y^{(h)}(t_i)\right) + \left(2E^{(h/2)}(t_i) + 4E^{(h/3)}(t_i)- E^{(h)}(t_i)\right)\right] $$

Step-by-step solution

Part (a): Derive a series expansion for the local truncation error

First, let's use the Taylor expansion to represent the forward Euler method: $$ y(t_{i+1}) = y(t_i) + hy'(t_i) + \frac{h^2}{2}y''(t_i) + \cdots $$ Now, let's subtract the forward Euler method from the Taylor expansion: $$ \begin{aligned} y(t_{i+1}) - (y(t_i) + hy'(t_i)) &= \frac{h^2}{2}y''(t_i) + \frac{h^3}{6}y'''(t_i) \\ &+ \frac{h^4}{24}y^{(4)}(t_i) + \frac{h^5}{120}y^{(5)}(t_i) + \mathcal{O}(h^6) \end{aligned} $$ The left side represents the local truncation error \(E(t_i)\) of the forward Euler method, which is given by the right side. Thus: $$ E(t_i) = \frac{h^2}{2}y''(t_i) + \frac{h^3}{6}y'''(t_i) + \frac{h^4}{24}y^{(4)}(t_i) + \frac{h^5}{120}y^{(5)}(t_i) + \mathcal{O}(h^6) $$

Part (b): Obtaining a second-order one-step method using extrapolation

Let's denote the forward Euler method for a step size of \(h\) by \(y^{(h)}\) and a step size of \(h/2\) by \(y^{(h/2)}\). Using the local truncation error derived in part (a), we can write the forward Euler method for the two step sizes: $$ \begin{aligned} y(t_{i+1}) &= y^{(h)}(t_i) + E^{(h)}(t_i) \\ y(t_{i+1}) &= y^{(h/2)}(t_i) + E^{(h/2)}(t_i) \end{aligned} $$ Now, let's manipulate these equations to eliminate the error term \(E^{(h)}(t_i)\) and obtain a second-order one-step method using extrapolation: $$ \begin{aligned} y(t_{i+1}) - E^{(h)}(t_i) &= y^{(h)}(t_i) \\ y(t_{i+1}) - E^{(h/2)}(t_i) &= y^{(h/2)}(t_i) \\ \end{aligned} $$ Then, we can multiply the second equation by 2 and subtract the first equation from the modified second equation: $$ \begin{aligned} 2\left[y(t_{i+1}) - E^{(h/2)}(t_i)\right] - \left[y(t_{i+1}) - E^{(h)}(t_i)\right] = y^{(h/2)}(t_i) \end{aligned} $$ Finally, solving for \(y(t_{i+1})\): $$ y(t_{i+1}) = \frac{1}{2}\left[2y^{(h/2)}(t_i) - y^{(h)}(t_i) + E^{(h)}(t_i) - E^{(h/2)}(t_i)\right] $$ This is the second-order one-step method obtained using extrapolation.

Part (c): Obtaining a third-order one-step method using extrapolation

Similar to part (b), we need to manipulate equations involving the forward Euler method with step sizes of \(h, h/2\), and \(h/3\) for this part. Let's denote the forward Euler method for a step size of \(h/3\) by \(y^{(h/3)}\). We'll also use the local truncation error derived in part (a) and write the forward Euler method for the three step sizes: $$ \begin{aligned} y(t_{i+1}) &= y^{(h)}(t_i) + E^{(h)}(t_i) \\ y(t_{i+1}) &= y^{(h/2)}(t_i) + E^{(h/2)}(t_i) \\ y(t_{i+1}) &= y^{(h/3)}(t_i) + E^{(h/3)}(t_i) \end{aligned} $$ To obtain a third-order one-step method using extrapolation, we must eliminate two error terms, \(E^{(h)}(t_i)\) and \(E^{(h/2)}(t_i)\), in the system of equations. We can do this by multiplying the second equation by 2, the third equation by 4, and summing them, then subtracting the first equation: $$ \begin{aligned} [2(y^{(h/2)}(t_i) + E^{(h/2)}(t_i)) + 4(y^{(h/3)}(t_i) + E^{(h/3)}(t_i))] - (y^{(h)}(t_i) + E^{(h)}(t_i)) = y(t_{i+1}) \end{aligned} $$ Now solve for \(y(t_{i+1})\): $$ \begin{aligned} y(t_{i+1}) = \frac{1}{8}\left[\left(2y^{(h/2)}(t_i) + 4y^{(h/3)}(t_i) - y^{(h)}(t_i)\right) + \left(2E^{(h/2)}(t_i) + 4E^{(h/3)}(t_i)- E^{(h)}(t_i)\right)\right] \end{aligned} $$ This is the third-order one-step method obtained using extrapolation.

Key concepts

Taylor Expansion

The Taylor expansion is a crucial mathematical tool in numerical methods that helps us approximate functions. This expansion expresses a function as an infinite sum of its derivatives at a single point. For example, using the Taylor series, we can approximate the value of a function at a point by taking into account the values and the derivatives of the function at another point nearby. The more derivatives we include, the better our approximation generally becomes. In numerical methods for solving differential equations, Taylor expansions are often used to derive methods like the Euler method, providing both the framework and insights into potential errors in approximations.

Local Truncation Error

When using numerical methods to approximate solutions to differential equations, we inevitably face errors. One of these is the local truncation error. This error occurs at each step of a numerical method due to the approximation technique itself, such as approximating with a truncated Taylor series instead of the full function. For example, in a simple Euler method, the local truncation error is given by the terms of the Taylor series that are omitted, which starts from the quadratic term onward. Understanding and managing this error is crucial when developing and using numerical methods because it accumulates over multiple steps and affects the accuracy of the final result.

Euler Method

The Euler method is one of the simplest numerical techniques to solve ordinary differential equations. It's a first-order method, which means it approximates solutions by considering only the first derivative (or linear term) of a Taylor expansion. To use the Euler method, we iteratively compute the next value using the formula:

• Current value:
• Derivative at current point:
• Step size:

The simplicity of the Euler method makes it easy to implement but also limits its accuracy, hence larger step sizes may lead to greater errors. It is a stepping stone to more accurate techniques that are developed by improving upon its fundamental approach.

Extrapolation

Extrapolation in numerical methods refers to refining solutions by carefully combining estimates at different scales. For instance, we might use values generated using different step sizes in the Euler method to eliminate lower-order error terms and achieve more accurate results. By cleverly subtracting scaled solutions for larger and smaller step sizes, we can "extrapolate" a more accurate estimate of the function's behavior. This process leverages the known behavior of errors in numerical solutions to effectively boost the method’s accuracy without the need for smaller step sizes directly in calculations, resulting in higher-order accuracy.

Higher Order Methods

Higher order methods in numerical computation refer to techniques that use additional derivatives or corrective steps to achieve more accurate solutions compared to simpler, lower-order methods like Euler's. By increasing the order, these methods reduce the impact of the local truncation error, leading to solutions that are closer to true values with fewer steps or larger intervals. These methods, such as Runge-Kutta techniques, take into account additional terms from the Taylor expansion, transforming our approach from a simpler method like Euler’s to one that effectively approximates with better accuracy and stability. Opting for higher order methods is particularly beneficial when dealing with complex systems requiring precise solutions.