I tend to think of Tolkien as someone who was knowledgeable about the natural world, and the kind of person who (like myself) makes a point of looking at the moon when possible and following its phases.
Certainly, it is known that the Lord of the Rings was interrupted for a prolonged spell in 1944-6 (when the Notion Club Papers were drafted) because of difficulty synchronizing the phases of the moon between different parts of the narrative. In fact he never quite managed to solve this problem - but it is a ather obscure matter, and doesn't mean much.
Yet Tolkien made the elementary mistake of recurrently describing his protagonists observing the New Moon Rising at night - when in fact the New Moon rises only during the day - after dawn, following the rising sun - indeed when it is newest the moon is invisible due to being lost in the light from the nearby sun.
(The time to see the New Moon is just after sunset, in the evening - when the New Moon is setting, not rising.)
The first comes from The Hobbit where Bard shoots Smaug the Dragon at the rising of the moon when the moon rose above the eastern shore and silvered [Smaug's] great wings... the waxing moon rose higher and higher.
Then we see something similar in the drafts of The Lord of the Rings published in the History of Middle Earth as The Return of the Shadow when Christopher Tolkien notes:
My father no doubt made this change on account of what he had said elsewhere about the Moon; for there was a waxing moon as the hobbits approached Weathertop, and it was 'nearly half-full' on the night of the attack: the attack was on 5 October...
and there could not be a full or nearly full Moon on 24 September, the night passed with the Elves in the Woody End... On that night it must have been almost New Moon. … But it is an odd and uncharacteristic aberration that my father envisaged a New Moon rising late at night in the East.