Question

(a) Show that the explicit trapezoidal method for \(y^{\prime}=f(y)\) is second order accurate. (b) Show that the explicit midpoint method for \(y^{\prime}=f(y)\) is second order accurate. [Hint: You need to compare derivatives of \(y\) with their expressions in terms of \(f\) and show that terms that are not \(\mathcal{O}\left(h^{2}\right)\) cancel in the expression for \(d_{i} .\) Use the identities \(y^{\prime}=f\) and \(\left.y^{\prime \prime}=\frac{\partial f}{\partial y} y^{\prime}=f_{y} f .\right]\)

Short answer

In summary, both the explicit trapezoidal method and the explicit midpoint method are second-order accurate for solving initial value problems involving first-order ordinary differential equations. This is because their local truncation errors at each step decrease quadratically as their mesh, or step size, is refined. In both methods, the terms that are not of the order \(\mathcal{O}\left(h^{2}\right)\) cancel each other, leaving only the second-order term dominating the error. This property makes these methods well-suited for various applications in numerical analysis and computational science.

Step-by-step solution

(a) Explicit Trapezoidal Method: Definition and Error Analysis

The explicit trapezoidal method for a first-order ordinary differential equation (ODE) \(y^{\prime} = f(y)\) is given by \[ y_{i+1} = y_{i} + \frac{h}{2} (f(y_i) + f(y_{i+1})), \] where \(y_{i}\) are the numerical approximations of the solution \(y(t)\) at the discrete points \(t_{i}\), and \(h\) is the step size. To analyze the local truncation error of the method, let's first derive an expression for the derivative \(y^{\prime}(t_{i+1})\). Using Taylor series expansion around \(t_{i}\), we have \[ y(t_{i+1}) = y(t_{i}) + h y^{\prime}(t_{i}) + \frac{h^2}{2} y^{\prime\prime}(t_{i}) + \mathcal{O}\left(h^{3}\right). \] Now, differentiate both sides of the ODE \(y^{\prime} = f(y)\) with respect to \(y\) and obtain \[ y^{\prime\prime} = \frac{\partial f}{\partial y}(y^{\prime}) = f_yf. \] Insert this into the Taylor series expansion above and rewrite in terms of \(f\): \[ y(t_{i+1}) = y(t_{i}) + hf(t_{i}) + \frac{h^2}{2} f_yf(t_{i}) + \mathcal{O}\left(h^{3}\right). \]

Calculate the Local Truncation Error

Now we subtract the numerical solution \(y_i\) on both sides of the Taylor series expansion and rearrange in terms of the local truncation error \(d_{i}\): \[ d_{i+1} = \frac{h}{2}(f(t_{i}) - f(t_{i+1})) + \frac{h^2}{2} f_yf(t_i) + \mathcal{O}\left(h^{3}\right). \] Since the terms involving the higher-order derivatives of \(y\) (which are third-order and beyond) are already represented within \(\mathcal{O}\left(h^{3}\right)\), we can focus on the two remaining terms to investigate the local truncation error. Observe that these terms are proportional to \(h^2\) and \(h^3\) respectively. So, we can conclude that the explicit trapezoidal method is second-order accurate.

(b) Explicit Midpoint Method: Definition and Error Analysis

The explicit midpoint method for a first-order ODE \(y^{\prime} = f(y)\) is given by \[ y_{i+1} = y_{i} + hf(y_{i} + \frac{1}{2}hf(y_{i})). \] To analyze the local truncation error of the method, let's derive an expression for the derivative \(y^{\prime}(t_{i+1})\) using a Taylor series expansion around \(t_{i}\) (as done in part a).

Calculate the Local Truncation Error

Subtracting the numerical solution \(y_i\) on both sides of the Taylor series expansion and rearranging in terms of the local truncation error \(d_{i}\), we get \[ d_{i+1} = y(t_{i+1}) - y_i - h \left[f(t_i) + \frac{1}{2}hf_yf(t_i)\right] + \mathcal{O}\left(h^{3}\right). \] Rearranging again, we obtain \[ d_{i+1} = h^2 \left[\frac{1}{2} f_yf(t_i) - \frac{1}{2}hf_yf(t_i)\right] + \mathcal{O}\left(h^{3}\right). \] Notice that the two remaining terms are again proportional to \(h^2\) and \(h^3\) respectively. Consequently, the explicit midpoint method is second-order accurate as well.

Key concepts

Explicit Trapezoidal Method

The explicit trapezoidal method is a numerical technique for approximating the solution of first-order ordinary differential equations (ODEs) of the form \(y^{\prime} = f(y)\). It is categorized as an explicit method because it directly provides the next value \(y_{i+1}\) based on the current value \(y_i\), requiring no iterative solution process at each step. The method formula is:\[y_{i+1} = y_i + \frac{h}{2} (f(y_i) + f(y_{i+1})),\]where \(h\) is the step size.

One significant trait of the explicit trapezoidal method is its second-order accuracy. This means that the error in approximating the solution decreases quadratically with the step size \(h\), according to the expression \(\mathcal{O}(h^2)\). Due to this higher order of accuracy, it often provides better approximations than first-order methods with the same step size.

In practical terms, utilizing this method involves balancing the step size for computational efficiency while maintaining acceptable error levels. By combining initial value conditions with the ODE's differential structure, this method helps systematically break down complex continuous problems into a series of discrete steps for approximate solutions.

Explicit Midpoint Method

The explicit midpoint method is another numerical method designed to solve first-order ODEs like \(y^{\prime} = f(y)\). It uses the concept of evaluating the function \(f\) at the midpoint of the interval to increase the accuracy of its results. Its formula is given by:

\[y_{i+1} = y_i + h f \left(y_i + \frac{1}{2}hf(y_i)\right).\]

The midpoint method's primary advantage is also its second-order accuracy. This implies that the approximation error is proportional to \(h^2\), making it more accurate compared to methods with only first-order accuracy like the Euler's method. The midpoint step ensures that you get a more refined estimation, effectively an average approximation over each interval.

The efficiency of this approach is evident when handling problems where reduced step sizes could lead to computational inefficiencies. For real applications, this method is practical for systems where moderate accuracy is required, such as in some engineering simulations.

Error Analysis

Error analysis in numerical methods is vital to understanding the performance of an algorithm. It provides insight into how closely a numerical solution approximates the true analytical solution of a differential equation. In the context of methods like explicit trapezoidal and midpoint, error analysis helps to assess their reliability.

We categorize errors into two main components:

• Truncation Error: Originates from the method's approximation, typically analyzed by comparing derivations like Taylor expansions to the true solution.
• Round-off Error: Results from numerical representations within computers, such as floating-point operations.

By studying how terms in the series expansion cancel each other out, as in the cases here where \(h^3\) terms are minimized, you can ensure that the remaining error terms are, at worst, \(\mathcal{O}(h^2)\).

This analysis is integral to algorithm development as it guides the step size selection for balancing computational load with desired accuracy. Particularly, second-order accurate methods, like those described, often offer the best compromise for many standard ODE problems.

Local Truncation Error

Local truncation error (LTE) refers specifically to the error introduced in one step of a numerical method when predicting the next value \(y_{i+1}\) from its current value \(y_i\). It effectively measures how much the numerical solution deviates from the true solution in a single step if previous steps were exact.

Formally, for a second-order accurate method, the LTE is \(\mathcal{O}(h^3)\), but since it accumulates over \(n\) steps, the global error becomes \(\mathcal{O}(h^2)\). This impact is crucial for understanding long-term behaviors of the numerical method used.

In analyses, higher-order derivatives (\(f_yf\)) in the expansion might be responsible for error terms. When these influence the LTE, one must consider them carefully during method design and application.

This concept is indispensable in numerical simulations across fields like physics, where dynamic systems modeled by differential equations heavily rely on accurate and manageable predictions over time.