Question

In this problem we discuss the global truncation error associated with the Euler method for the initial value problem \(y^{\prime}=f(t, y), y\left(t_{0}\right)=y_{0}\). Assuming that the functions \(f\) and \(f_{y}\) are continuous in a region \(R\) of the \(t y\) -plane that includes the point \(\left(t_{0}, y_{0}\right),\) it can be shown that there exists a constant \(L\) such that \(|f(t, y)-f(t, \tilde{y}|

Short answer

**Question**: Derive inequalities and error bounds for the global truncation error associated with the Euler method for an initial value problem with a given function \(f(t, y)\) and its partial derivatives \(f_y\) and \(f_t\). Prove that \((1+hL)^n \leq e^{nhL}\) and provide another form for the error bound illustrating its relation to the error bounds obtained through an intuitive argument. **Answer**: By analyzing the equation for the Euler method error, we can derive the following inequality for global truncation error: $$ \left|E_{n+1}\right| \leq (1+hL)\left|E_n\right| + \frac{1}{2}h^2\left|\phi^{\prime\prime}(\bar{t}_n)\right| \leq \alpha\left|E_n\right|+\beta h^{2} $$ Where \(\alpha = 1 + hL\) and \(\beta = \max \left|\phi^{\prime\prime}(t)\right| / 2\) on \(t_0 \leq t \leq t_n\). We can then derive the following relationship for the error bound: $$ \left|E_{n}\right| \leq \frac{(1+hL)^{n}-1}{L} \beta h $$ Now, we prove that \((1+hL)^{n} \leq e^{nhL}\) : $$ \left(1+hL\right)^n \leq \left(e^{hL}\right)^n = e^{nhL} $$ Using this result, we can rewrite the error bound as: $$ \left|E_{n}\right| \leq \frac{e^{nhL}-1}{L} \beta h = \frac{e^{(\zeta_{n}-t_0)L}-1}{L} \beta h $$ When \(nhL = (\bar{t} - t_0)L\) is small, this error bound is approximately \(nh^2\beta = (\bar{t} - t_0) \beta h\), which was obtained in Eq. (24) by an intuitive argument.

Step-by-step solution

(Part a) Derivation of Inequality and Error Bounds

First, let's write down Eq. (20) which is given by: $$ E_{n+1} = E_n + hf(t_n, \phi(t_n)) -hf(t_n, y_n) + \frac{1}{2}h^2\phi^{\prime\prime}(\bar{t}_n) $$ Now, we will analyze the absolute of this equation: $$ \left|E_{n+1}\right| \leq \left|E_n\right| + h\left|f(t_n, \phi(t_n)) - f(t_n, y_n)\right| + \frac{1}{2}h^2\left|\phi^{\prime\prime}(\bar{t}_n)\right| $$ Since \(|f(t, y) - f(t, \tilde{y})| < L|y - \tilde{y}|\), we can use this inequality in the above equation: $$ \left|E_{n+1}\right| \leq \left|E_n\right| + hL\left|E_n\right| + \frac{1}{2}h^2\left|\phi^{\prime\prime}(\bar{t}_n)\right| $$ Rearranging terms, we get: $$ \left|E_{n+1}\right| \leq (1+hL)\left|E_n\right| + \frac{1}{2}h^2\left|\phi^{\prime\prime}(\bar{t}_n)\right| \leq \alpha\left|E_n\right|+\beta h^{2} $$ Where \(\alpha = 1 + hL\) and \(\beta = \max \left|\phi^{\prime\prime}(t)\right| / 2\) on \(t_0 \leq t \leq t_n\).

(Part b) Error Bound with Exponential Factor

Given without proof that if \(E_{0}=0\), and if \(\left|E_{n}\right|\) satisfies Eq. (i), then \(\left|E_{n}\right| \leq \beta h^{2}\left(\alpha^{n}-1\right) /(\alpha-1)\) for \(\alpha \neq 1\), we can derive the following relationship: $$ \left|E_{n}\right| \leq \frac{(1+hL)^{n}-1}{L} \beta h $$

(Part c) Upper Bound of Exponential Factor

Now, let's prove that \((1+hL)^{n} \leq e^{nhL}\): Using the exponential property \(e^x \geq 1+x\) for any real number \(x\): $$ \left(1+hL\right)^n \leq \left(e^{hL}\right)^n = e^{nhL} $$ Thus, we can rewrite the error bound as: $$ \left|E_{n}\right| \leq \frac{e^{nhL}-1}{L} \beta h = \frac{e^{(\zeta_{n}-t_0)L}-1}{L} \beta h $$ For a fixed point \(\bar{t}=t_0+nh\) (i.e., \(nh\) is constant and \(h = (\bar{t}-t_0) / n\)), this error bound is of the form of a constant times \(h\) and approaches zero as \(h \rightarrow 0\). Notice that when \(nhL = (\bar{t} - t_0)L\) is small, the right side of the equation is approximately \(nh^2\beta = (\bar{t} - t_0) \beta h\), which was obtained in Eq. (24) by an intuitive argument.

Key concepts

Global Truncation Error

In numerical analysis, when we use the Euler method to solve differential equations, we're often concerned about the accuracy of our solutions. A key aspect of this is the global truncation error. This error helps us understand how much our numerical solution deviates from the true solution over an interval. It accumulates over each step in the Euler method and provides insights into how the errors in each step affect the final solution.

The global truncation error can be expressed as a bound that increases with the distance from the initial point. In typical problems, it depends on the step size \( h \), a constant \( L \) related to the function's derivatives, and the number of steps \( n \). The equation \(|E_{n}| \leq \frac{(1+hL)^{n}-1}{L} \beta h\) gives a way to estimate this error, showing that smaller step sizes \( h \) generally lead to less accumulated error over time. By managing step size and understanding this error, we can achieve a balance between computational efficiency and accuracy.

Initial Value Problem

When working with differential equations, we often encounter the initial value problem (IVP). An IVP specifies a differential equation along with a condition that the solution must satisfy at a specific point - typically given as \( y(t_0) = y_0 \). This serves as the starting point for the numerical solution.

The Euler method, for example, requires us to start at \( t_0 \) with the initial value \( y_0 \) and then use iterative steps to approximate the solution over a time interval. This initial value anchors our calculations and ensures that the numerical approximation follows a defined path that mimics the real dynamical system described by the differential equation. Understanding the role of initial conditions is crucial as they heavily influence the trajectory of the solution.

Error Bounds

Error bounds in numerical analysis denote limits within which the true value of a calculation is expected to lie. These are essential to validate the viability of numerical methods like the Euler method.

An error bound gives us a worst-case scenario estimate, ensuring confidence that the real error will not exceed this bound under specified conditions. In particular for Euler methods, we use bounds involving step size \( h \), the number of steps \( n \), and constants related to the nature of the functions involved, such as \( L \) and \( \beta \).

In practical use, recognizing these bounds helps in adjusting parameters to optimize the precision of approximations and manage computational resources efficiently.

Numerical Analysis

Numerical analysis involves approximating mathematical procedures numerically and is a crucial field for solving problems that do not have exact analytic answers.

This can include differential equations, where exact solutions are rare or too complicated to compute for practical applications. In this context, numerical methods like the Euler method come into play. By using steps of a fixed size, they enable iterating towards a solution, providing a valuable toolkit for engineers and scientists exploring complex systems.

One primary focus in numerical analysis is understanding and controlling errors. Through techniques like analyzing global truncation errors and formulating error bounds, the field ensures that the approximations remain useful despite inherent inaccuracies. Thus, numerical analysis not only offers methods to approximate solutions but also equips us with the means to assess and refine those methods.