Amplify Arduino Due DAC output
PWM

If you can tolerate a bit of ripple on the output and precision isn't critical, and since the load is an inductor, the simplest solution would be a PWM brushed motor controller like this one, or any other as long as it supports the required voltage and current.

The h-bridge principle is explained here. You'd control this with a PWM output from your microcontroller, and another pin for direction/polarity.

Advantages: simple, cheap, efficient, quick transient response, needs only one positive supply, no analog signal conditioning.

Drawbacks: the output is the PWM value multiplied by supply voltage, so its accuracy depends on the supply, and of course it will have ripple, so you'll have to calculate the current ripple in your inductor according to the PWM frequency to make sure that it will do what you want.

Analog

The link you give in the question will translate DAC output voltage to 0-3.3V but you still need to amplify it to 15V, with an amp that can deliver enough current, and if you want to be able to reverse the polarity while using a single power supply, then it will have to be a bridged amplifier.

Since your coil's DC resistance is 5 ohms, circuits designed for car audio (12V and 4 ohm loudspeakers) should be a nice fit, something like TDA7384, STPA008 etc. 

Or you can use a current booster on an opamp, or even design a discrete amp, but if you want to keep it simple, if I were you I'd just use a very cheap class-D module with TPA3116. This will handle the current you need. You will have to short the input DC coupling cap to pass your DC signal though. Since these things cost under $10, it's worth a try.