Question

To draw a circle of radius \(r\) on a graphics screen, one may proceed to evaluate pairs of values \(x=r \cos (\theta), y=r \sin (\theta)\) for a succession of values \(\theta\). But this is computationally expensive. A cheaper method may be obtained by considering the \(\mathrm{ODE}\) $$ \begin{array}{lc} \dot{x}=-y, & x(0)=r, \\ \dot{y}=x, & y(0)=0 \end{array} $$ where \(\dot{x}=\frac{d x}{d \theta}\), and approximating this using a simple discretization method. However, care must be taken to ensure that the obtained approximate solution looks right, i.e., that the approximate curve closes rather than spirals. Carry out this integration using a uniform step size \(h=.02\) for \(0 \leq \theta \leq 120\), applying forward Euler, backward Euler, and the implicit trapezoidal method. Determine if the solution spirals in, spirals out, or forms an approximate circle as desired. Explain the observed results. [Hint: This has to do with a certain invariant function of \(x\) and \(y\), rather than with the accuracy order of the methods.]

Short answer

Question: Given the parametric equations of a circle with radius \(r\), we need to approximate the curves obtained by using the forward Euler method, backward Euler method, and implicit trapezoidal method on the ODE systems. For each case, determine whether the curve spirals in, spirals out, or forms an approximate circle, and explain the results. Solution: Once the three numerical integration methods are implemented and the curve for each method is plotted, analyze whether it spirals in, spirals out, or forms an approximate circle.

Step-by-step solution

ODE System

The given ODE system is: $$ \begin{cases} \dot{x}=-y \\ \dot{y}=x \\ \end{cases}, $$ with initial conditions \(x(0)=r\) and \(y(0)=0\). We want to approximate the solutions for \(0 \leq \theta \leq 120\) with uniform step size \(h=0.02\).

Forward Euler Method

To compute the Forward Euler method for this ODE system, we follow these steps at each step \(k\): 1. Update \(x_{k+1} = x_k - h y_k\). 2. Update \(y_{k+1} = y_k + h x_k\). Repeat these steps for \(0 \leq \theta \leq 120\) with \(h=0.02\).

Backward Euler Method

To compute the Backward Euler method for this ODE system, we follow these steps at each step \(k\): 1. Solve \(x_{k+1} = x_k - h y_{k+1}\) for \(x_{k+1}\). 2. Solve \(y_{k+1} = y_k + h x_{k+1}\) for \(y_{k+1}\). Repeat these steps for \(0 \leq \theta \leq 120\) with \(h=0.02\).

Implicit Trapezoidal Method

To compute the Implicit Trapezoidal method for this ODE system, we follow these steps at each step \(k\): 1. Solve \(x_{k+1} = x_k - \frac{h}{2}(y_k + y_{k+1})\) for \(x_{k+1}\). 2. Solve \(y_{k+1} = y_k + \frac{h}{2}(x_k + x_{k+1})\) for \(y_{k+1}\). Repeat these steps for \(0 \leq \theta \leq 120\) with \(h=0.02\).

Comparison of Methods

After implementing and computing the three numerical integration methods, plot the results to visualize the behavior of the curve for each method. Furthermore, determine whether the generated curve spirals in, spirals out, or forms an approximate circle, and explain the observed results considering an invariant function of \(x\) and \(y\).

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are fundamental in modeling various phenomena in mathematics and science. An ODE involves a function of one or more variables and its derivatives. The key point about ODEs is that they provide relationships involving rates of change. For example, in our exercise, we have the equations \( \dot{x} = -y \) and \( \dot{y} = x \), where the dots denote derivatives with respect to \( \theta \). These equations are related to the motion of a point on a circle.
To solve an ODE for practical use, such as drawing a circle, numerical methods are employed. These methods approximate the values of the function at discrete points within the interval of interest. Solving ODEs accurately is crucial since they tell us how certain variables depend on each other and evolve over time.
In the specific context of the exercise, we're using ODEs to simulate the movement of a point along the circumference of a circle through a series of iterative calculations.

Euler's Method

Euler's Method is one of the simplest and most well-known numerical techniques for approximating solutions to ODEs. This method involves a straightforward iteration process to get an approximate solution. In essence, it uses tangent line approximations to iteratively arrive at the solution point.
For example, given the system in the exercise, the Forward Euler method updates \( x \) and \( y \) iteratively with \( x_{k+1} = x_k - h y_k \) and \( y_{k+1} = y_k + h x_k \), where \( h \) is the step size. It's easy to implement but may not always be accurate because it utilizes only the slope at the beginning of the step interval.
Moreover, while easy to understand and implement, one downside of Euler's method is that it may not preserve important properties like energy or invariants in the system, which can lead to errors accumulating in a spiral when modeling circular motion.

Implicit Methods

Implicit Methods, such as the Backward Euler method, are more advanced techniques for solving ODEs than their explicit counterparts. These methods differ because instead of using information from the current or past state, they incorporate future information in the step calculation. This typically requires solving algebraic equations at each step, which can be computationally more intensive.
In the context of the exercise, the Backward Euler method involves equations \( x_{k+1} = x_k - h y_{k+1} \) and \( y_{k+1} = y_k + h x_{k+1} \). You notice that \( y_{k+1} \) and \( x_{k+1} \) are defined in terms of each other, making it implicit. These methods are stable for larger step sizes, which can lead to more accurate results when simulating the ODE over a long interval.
Implied in the details of implicit methods is the advantage of stability, especially when tackling stiff ODEs, ensuring that the numerical solution does not spiral inadvertently, thus maintaining closer adherence to circular motion in our circle drawing problem.

Circle Drawing Algorithm

The Circle Drawing Algorithm is a computational technique aimed at representing a circle on a digital screen efficiently. Using direct trigonometric calculations for every angle \( \theta \) such as \( x=r \cos(\theta) \) and \( y=r \sin(\theta) \) can be computationally expensive due to the calculations of trigonometric functions.
In our exercise, the algorithm involves solving an ODE system as an alternative to simply approximating \( x \) and \( y \) values directly. This method reduces computational load by using numeric integration techniques like Euler’s Method or Implicit Methods, which focus on updating the values iteratively based on differential changes. This not only speeds up the process but also maintains accuracy for large spans of \( \theta \).
However, special care is needed to ensure the final set of points form a closed loop rather than spiraling inward or outward, which can happen if numerical errors accumulate. So, balancing between computational efficiency and accuracy is key when implementing a circle drawing algorithm.