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Partial Difference Equations by Sui Sun Cheng? A good start (in the context of reading the classics) is the classic paper by Courant-Friedrichs-Lewy. You can find Also useful is a list with more books.

For the non-partial case, I must admit I have a soft spot for Milne-Thompson's book.

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Partial Difference Equations by Sui Sun Cheng? A good start (in the context of reading the classics) is the classic by Courant-Friedrichs-Lewy. You can find more books.

For the non-partial case, I must admit I have a soft spot for Milne-Thompson's book.