﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:03.00,Default,,0000,0000,0000,,Now, let's have a look at the zoo of methods that we've seen so far Dialogue: 0,0:00:03.00,0:00:06.00,Default,,0000,0000,0000,,and then let's add one further method to that collection. Dialogue: 0,0:00:06.00,0:00:10.00,Default,,0000,0000,0000,,The forward Euler method work by starting from the current value Dialogue: 0,0:00:10.00,0:00:17.00,Default,,0000,0000,0000,,and then incrementing that by step size h times the rate of change at the current point. Dialogue: 0,0:00:17.00,0:00:20.00,Default,,0000,0000,0000,,I'm writing f for the rate of change here. Dialogue: 0,0:00:20.00,0:00:24.00,Default,,0000,0000,0000,,Heun method also known as the improved or the modified Euler method Dialogue: 0,0:00:24.00,0:00:31.00,Default,,0000,0000,0000,,advances by the time step times the average of the rate of change at the beginning Dialogue: 0,0:00:31.00,0:00:36.00,Default,,0000,0000,0000,,plus the rate of change at the position predicted by the forward Euler method. Dialogue: 0,0:00:36.00,0:00:42.00,Default,,0000,0000,0000,,With the forward Euler method, the error of a numerical solution grows linearly with step size. Dialogue: 0,0:00:42.00,0:00:48.00,Default,,0000,0000,0000,,If you double the step size, the global error also doubles approximately at least. Dialogue: 0,0:00:48.00,0:00:57.00,Default,,0000,0000,0000,,The forward Euler method is a method of order 1. This function, the error grows like h¹. Dialogue: 0,0:00:57.00,0:01:03.00,Default,,0000,0000,0000,,Heun method, however, is the solver of order 2. The error grows like h squared. Dialogue: 0,0:01:03.00,0:01:10.00,Default,,0000,0000,0000,,If you take half the step size, the error is shrinking to one quarter, which is much more efficient. Dialogue: 0,0:01:10.00,0:01:15.00,Default,,0000,0000,0000,,For the backward Euler method, you advance by the rate of change at the new position, Dialogue: 0,0:01:15.00,0:01:18.00,Default,,0000,0000,0000,,which makes this equation difficult--it's an implicit equation. Dialogue: 0,0:01:18.00,0:01:23.00,Default,,0000,0000,0000,,To complete our zoo of methods, we add a method that looks like Heun method Dialogue: 0,0:01:23.00,0:01:27.00,Default,,0000,0000,0000,,but is implicit like the backward Euler method. Dialogue: 0,0:01:27.00,0:01:32.00,Default,,0000,0000,0000,,This new rule is called the trapezoidal rule. Like Heun method, it's a method of order 2. Dialogue: 0,0:01:32.00,0:01:35.00,Default,,0000,0000,0000,,And if you compare these equations, they look pretty similar. Dialogue: 0,0:01:35.00,0:01:40.00,Default,,0000,0000,0000,,The trapezoidal rule advances by the average of the rate of change at the beginning Dialogue: 0,0:01:40.00,9:59:59.99,Default,,0000,0000,0000,,and the rate of change at the end, which means that this is an implicit equation.