90 lines
4.1 KiB
Markdown
90 lines
4.1 KiB
Markdown
---
|
|
title: Euler method
|
|
id: 59880443fb36441083c6c20e
|
|
challengeType: 5
|
|
---
|
|
|
|
## Description
|
|
<section id='description'>
|
|
<p>Euler's method numerically approximates solutions of first-order ordinary differential equations (ODEs) with a given initial value. It is an explicit method for solving initial value problems (IVPs), as described in <a href="https://en.wikipedia.org/wiki/Euler method" title="wp: Euler method">the wikipedia page</a>.</p><p>The ODE has to be provided in the following form:</p><p>:: <big>$\frac{dy(t)}{dt} = f(t,y(t))$</big></p><p>with an initial value</p><p>:: <big>$y(t_0) = y_0$</big></p><p>To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:</p><p>:: <big>$\frac{dy(t)}{dt} \approx \frac{y(t+h)-y(t)}{h}$</big></p><p>then solve for $y(t+h)$:</p><p>:: <big>$y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$</big></p><p>which is the same as</p><p>:: <big>$y(t+h) \approx y(t) + h \, f(t,y(t))$</big></p><p>The iterative solution rule is then:</p><p>:: <big>$y_{n+1} = y_n + h \, f(t_n, y_n)$</big></p><p>where <big>$h$</big> is the step size, the most relevant parameter for accuracy of the solution. A smaller step size increases accuracy but also the computation cost, so it has always has to be hand-picked according to the problem at hand.</p>
|
|
<p>Example: Newton's Cooling Law</p><p>Newton's cooling law describes how an object of initial temperature <big>$T(t_0) = T_0$</big> cools down in an environment of temperature <big>$T_R$</big>:</p><p>:: <big>$\frac{dT(t)}{dt} = -k \, \Delta T$</big></p>
|
|
<p>or</p>
|
|
<p>:: <big>$\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$</big></p>
|
|
<p>It says that the cooling rate <big>$\frac{dT(t)}{dt}$</big> of the object is proportional to the current temperature difference <big>$\Delta T = (T(t) - T_R)$</big> to the surrounding environment.</p><p>The analytical solution, which we will compare to the numerical approximation, is</p>
|
|
<p>:: <big>$T(t) = T_R + (T_0 - T_R) \; e^{-k t}$</big></p>
|
|
Task:
|
|
<p>Implement a routine of Euler's method and then to use it to solve the given example of Newton's cooling law with it for three different step sizes of:</p>
|
|
<p>::* 2 s</p>
|
|
<p>::* 5 s and </p>
|
|
<p>::* 10 s </p>
|
|
<p>and to compare with the analytical solution.</p>
|
|
Initial values:
|
|
<p>::* initial temperature <big>$T_0$</big> shall be 100 °C</p>
|
|
<p>::* room temperature <big>$T_R$</big> shall be 20 °C</p>
|
|
<p>::* cooling constant <big>$k$</big> shall be 0.07 </p>
|
|
<p>::* time interval to calculate shall be from 0 s ──► 100 s</p>
|
|
</section>
|
|
|
|
## Instructions
|
|
<section id='instructions'>
|
|
|
|
</section>
|
|
|
|
## Tests
|
|
<section id='tests'>
|
|
|
|
```yml
|
|
tests:
|
|
- text: <code>eulersMethod</code> is a function.
|
|
testString: 'assert(typeof eulersMethod === "function", "<code>eulersMethod</code> is a function.");'
|
|
- text: '<code>eulersMethod(0, 100, 100, 10)</code> should return a number.'
|
|
testString: 'assert(typeof eulersMethod(0, 100, 100, 10) === "number", "<code>eulersMethod(0, 100, 100, 10)</code> should return a number.");'
|
|
- text: '<code>eulersMethod(0, 100, 100, 10)</code> should return 20.0424631833732.'
|
|
testString: 'assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732, "<code>eulersMethod(0, 100, 100, 10)</code> should return 20.0424631833732.");'
|
|
- text: '<code>eulersMethod(0, 100, 100, 10)</code> should return 20.01449963666907.'
|
|
testString: 'assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907, "<code>eulersMethod(0, 100, 100, 10)</code> should return 20.01449963666907.");'
|
|
- text: '<code>eulersMethod(0, 100, 100, 10)</code> should return 20.000472392.'
|
|
testString: 'assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392, "<code>eulersMethod(0, 100, 100, 10)</code> should return 20.000472392.");'
|
|
|
|
```
|
|
|
|
</section>
|
|
|
|
## Challenge Seed
|
|
<section id='challengeSeed'>
|
|
|
|
<div id='js-seed'>
|
|
|
|
```js
|
|
function eulersMethod (x1, y1, x2, h) {
|
|
// Good luck!
|
|
}
|
|
```
|
|
|
|
</div>
|
|
|
|
|
|
|
|
</section>
|
|
|
|
## Solution
|
|
<section id='solution'>
|
|
|
|
|
|
```js
|
|
function eulersMethod(x1, y1, x2, h) {
|
|
let x = x1;
|
|
let y = y1;
|
|
|
|
while ((x < x2 && x1 < x2) || (x > x2 && x1 > x2)) {
|
|
y += h * (-0.07 * (y - 20));
|
|
x += h;
|
|
}
|
|
|
|
return y;
|
|
}
|
|
|
|
```
|
|
|
|
</section>
|