If the faster strategy involving the barrage of grenades doesn't work for you, try this longer version. It is also covered in the Rutland video.
Begin with pistols to conserve ammo. Run, jump and roll to stay out of Rutland's way as much as possible. You may want to try and stay on the opposite side of the arena from him, circling around as he moves to keep him opposite Lara.
Don't let him get Lara up against a wall. When he throws a grenade leap away from it, as shown above, and keep shooting at him whenever you get an opportunity.
Once he's lost about a third of his health, he'll leap up onto one of the ledges to heal. As soon as he does this for the first time, you will be able to target the red crystals on the underside of each ledge. Do this by holding Interact, as you would when shooting an environmental hazard. Quickly shoot one of the crystals to reveal a shiny grapple point.
Immediately holster your weapons, grapple the ledge and press Interact to pull it down.
By the time you finish, Rutland should be back at full health and ready to continue fighting.
You can sometimes gain a few extra seconds by pulling down the ledge on which he's standing.
When it falls, he'll take a moment to get up, during which you can continue to fire.
You can either try and shoot all the crystals before pulling down the ledges or concentrate on completely destroying one ledge at a time. If you're agile, you may be able to dodge Rutland's attacks while working on the ledges, but I found it easier to shoot him with pistols until he retreats. Then use that time to destroy as many ledges as possible.
Once all four ledges are gone, Rutland can no longer regenerate, so you can pull out the heavy artillery. Continue to attack while dodging his grenades and avoiding his slashing attacks.
Sometimes, when Rutland loses a third of his health, he'll run toward the wall, as though getting ready to jump onto a ledge. If this happens, start lobbing grenades at him, as shown above. You may be able to finish him off this way. Otherwise, just keep plugging away until he finally falls.
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