I have myself only glanced through it, but I believe the book An Introduction to Dynamic Meteorology by James R. Holton (4th edition 2004) is a standard reference in the domain. It's more on the physical than the mathematical side of things, but it's fairly self-contained, and does have a chapter on numerical methods. And it has accompanying Matlab files that are used as exercises to illustrate how various concepts are simulated in software.
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