1 00:00:00,000 --> 00:00:03,000 Now, let's have a look at the zoo of methods that we've seen so far 2 00:00:03,000 --> 00:00:06,000 and then let's add one further method to that collection. 3 00:00:06,000 --> 00:00:10,000 The forward Euler method work by starting from the current value 4 00:00:10,000 --> 00:00:17,000 and then incrementing that by step size h times the rate of change at the current point. 5 00:00:17,000 --> 00:00:20,000 I'm writing f for the rate of change here. 6 00:00:20,000 --> 00:00:24,000 Heun method also known as the improved or the modified Euler method 7 00:00:24,000 --> 00:00:31,000 advances by the time step times the average of the rate of change at the beginning 8 00:00:31,000 --> 00:00:36,000 plus the rate of change at the position predicted by the forward Euler method. 9 00:00:36,000 --> 00:00:42,000 With the forward Euler method, the error of a numerical solution grows linearly with step size. 10 00:00:42,000 --> 00:00:48,000 If you double the step size, the global error also doubles approximately at least. 11 00:00:48,000 --> 00:00:57,000 The forward Euler method is a method of order 1. This function, the error grows like hÂ¹. 12 00:00:57,000 --> 00:01:03,000 Heun method, however, is the solver of order 2. The error grows like h squared. 13 00:01:03,000 --> 00:01:10,000 If you take half the step size, the error is shrinking to one quarter, which is much more efficient. 14 00:01:10,000 --> 00:01:15,000 For the backward Euler method, you advance by the rate of change at the new position, 15 00:01:15,000 --> 00:01:18,000 which makes this equation difficult--it's an implicit equation. 16 00:01:18,000 --> 00:01:23,000 To complete our zoo of methods, we add a method that looks like Heun method 17 00:01:23,000 --> 00:01:27,000 but is implicit like the backward Euler method. 18 00:01:27,000 --> 00:01:32,000 This new rule is called the trapezoidal rule. Like Heun method, it's a method of order 2. 19 00:01:32,000 --> 00:01:35,000 And if you compare these equations, they look pretty similar. 20 00:01:35,000 --> 00:01:40,000 The trapezoidal rule advances by the average of the rate of change at the beginning 21 00:01:40,000 --> 99:59:59,999 and the rate of change at the end, which means that this is an implicit equation.