Question

Carry out one step of the Euler method and of the improved Euler method using the step size \(h=0.1 .\) Suppose that a local truncation error no greater than 0.0025 is required. Estimate the step size that is needed for the Euler method to satisfy this requirement at the first step. $$ y^{\prime}=0.5-t+2 y, \quad y(0)=1 $$

Short answer

Based on the step by step solution provided, 1. After one step, the approximate value using the Euler method is \(y_1 = 1.25\). 2. After one step, the approximate value using the improved Euler method is \(y_1 \approx 1.2475\). 3. To satisfy the local truncation error requirement of 0.0025 at the first step, a step size of approximately \(h_{new} \approx 0.1581\) is needed for the Euler method.

Step-by-step solution

Euler method (one step)

To perform one step of the Euler method, we use the formula: $$ y_{n+1} = y_n + h \cdot f(t_n, y_n) $$ With the given initial values, \(y_0 = 1\) and \(t_0 = 0\), we can calculate \(y_1\): $$ y_{1} = y_0 + h \cdot f(t_0, y_0) = 1 + 0.1 \cdot (0.5 - 0 + 2\cdot 1) = 1 + 0.1 \cdot 2.5 = 1.25 $$ Thus, after the first step, the approximate value is \(y_1 = 1.25\).

Improved Euler method (one step)

To perform one step of the improved Euler method, we first calculate an intermediate value, then use it to calculate a new value for \(y\): $$ k_1 = f(t_0, y_0) $$ $$ k_2 = f(t_0 + h, y_0 + hk_1) $$ $$ y_{n+1} = y_n + \frac{1}{2}h(k_1 + k_2) $$ With our initial values, we first calculate \(k_1\) and \(k_2\): $$ k_{1} = f(0, 1) = 0.5 - 0 + 2 \cdot 1 = 2.5 $$ $$ k_{2} = f(0.1, 1 + 0.1\cdot 2.5) = 0.5 - 0.1 + 2 \cdot 1.25 = 2.45 $$ Now, we can calculate \(y_1\): $$ y_{1} = y_0 + \frac{1}{2} \cdot 0.1 (k_1 + k_2) = 1 + 0.05 \cdot (2.5 + 2.45) = 1 + 0.05 \cdot 4.95 \approx 1.2475 $$ Hence, after the first step, the approximate value is \(y_1 \approx 1.2475\) using the improved Euler method.

Estimate step size for Euler method with local truncation error requirement

Now we need to estimate the step size needed for the Euler method to satisfy the local truncation error requirement of 0.0025 at the first step. The Euler method's local truncation error is proportional to the square of the step size, i.e., $$ E = O(h^2) $$ To find the required step size, we can first calculate the ratio between the desired local truncation error and the given step size squared: $$ R = \frac{0.0025}{0.1^2} = 2.5 $$ By taking the square root of this ratio, we can estimate the required step size: $$ h_{new} = \sqrt{R} \cdot h = \sqrt{2.5} \cdot 0.1 \approx \0.1581 $$ Thus, to satisfy the local truncation error requirement of 0.0025 at the first step, we need to use a step size of approximately \(h_{new} \approx 0.1581\).

Key concepts

Improved Euler Method

Euler's method is a simple numerical procedure to find approximate solutions to differential equations, but it can be improved for better accuracy. The improved Euler method, also known as the Modified Euler or Heun's method, adds an extra step to help refine the approximation.
Instead of just calculating the slope at the initial point, the improved Euler method takes an additional midpoint approximation into consideration. Here’s how it works:

• Step 1: Calculate the first slope estimate, called \(k_1\), at the initial point \((t_n, y_n)\).
• Step 2: Use \(k_1\) to estimate an intermediate point \((t_n + h, y_n + hk_1)\).
• Step 3: Calculate a second slope estimate, \(k_2\), at this intermediate point.
• Step 4: Finally, use both \(k_1\) and \(k_2\) to find the next point \(y_{n+1}\) with the formula \(y_{n+1} = y_n + \frac{1}{2}h(k_1 + k_2)\)

This additional step of averaging the slopes \(k_1\) and \(k_2\) gives the improved Euler method a good balance of accuracy and computational simplicity, making it a better choice when compared to the simple Euler method.

Local Truncation Error

Every numerical method for differential equations introduces some error at each step, because we're approximating a continuous process using discrete steps. This error is known as the local truncation error (LTE). Calculating LTE helps in understanding how much the numerical solution differs from the exact solution locally, i.e., from one step to the next.

For the Euler method, the local truncation error depends on the square of the step size \(h\). This means that if you reduce the step size by half, the error decreases approximately by a factor of four. This relationship is mathematically represented as:
\[ E \approx C \cdot h^2 \]
where \( E \) is the truncation error, and \( C \) is a constant that depends on the function being solved.
To maintain a controlled level of accuracy, like a local truncation error under a certain value, adjusting the step size \(h\) is crucial. For the Euler method, if a smaller local truncation error is required, the step size must also be decreased, which may require more computation.

Differential Equations

Differential equations are mathematical expressions that describe the relationship between a function and its derivatives. They play a fundamental role in modeling the dynamics of systems that change continuously over time, such as motion, growth processes, and circuits.

The general form of an ordinary differential equation (ODE) is:
\[ y' = f(t, y) \]
where \(y'\) is the derivative of the unknown function \(y\) with respect to time or another variable \(t\). The function \(f(t, y)\) describes how \(y\) changes with \(t\).
Solving differential equations is a key task in the field of science and engineering. Some equations have analytical solutions, which can be expressed in closed form, using well-known functions. However, many real-world problems can only be solved numerically, through methods like Euler's method, the improved Euler method, or other more advanced techniques.

By understanding the basics of differential equations, you gain the ability to describe and predict a wide variety of dynamic behaviors in real-world systems.