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Now, let's have a look at the zoo of methods that we've seen so far

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and then let's add one further method to that collection.

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The forward Euler method work by starting from the current value

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and then incrementing that by step size h times the rate of change at the current point.

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I'm writing f for the rate of change here.

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Heun method also known as the improved or the modified Euler method

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advances by the time step times the average of the rate of change at the beginning

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plus the rate of change at the position predicted by the forward Euler method.

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With the forward Euler method, the error of a numerical solution grows linearly with step size.

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If you double the step size, the global error also doubles approximately at least.

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The forward Euler method is a method of order 1. This function, the error grows like h¹.

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Heun method, however, is the solver of order 2. The error grows like h squared.

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If you take half the step size, the error is shrinking to one quarter, which is much more efficient.

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For the backward Euler method, you advance by the rate of change at the new position,

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which makes this equation difficult--it's an implicit equation.

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To complete our zoo of methods, we add a method that looks like Heun method

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but is implicit like the backward Euler method.

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This new rule is called the trapezoidal rule. Like Heun method, it's a method of order 2.

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And if you compare these equations, they look pretty similar.

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The trapezoidal rule advances by the average of the rate of change at the beginning

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and the rate of change at the end, which means that this is an implicit equation.