I very recently started to read up about magnetohydrodynamics (MHD). While I have experience in the fluid part (both theory and numerics), my knowledge about the magneto part is very limited.
At the moment, I am working through the book by Davidson which is great for learning about physics. I decided that a good first step will be to write my own simple code solving the induction equation
\begin{equation} B_t = \nabla \times \left( \mathbf{u} \times B \right). \end{equation}
The problem is I do not know how a specific choice of numerical method will perform for this problem nor how good test cases would look like.
Therefore, I am looking for a good introductory book or script on numerical methods for MHD. Ideally, I hope to find something that is similar to the book by Durran for geophysical fluid dynamics (GFD) - a thorough introduction to different numerical methods used in the field and analysis of their performance from simple to complex benchmarks problems.
Addendum: To clarify my question a little bit, I am not looking for general introductions into methods that are used in MHD (finite differences, specific integration methods, finite elements, etc). Rather, I am looking for a book that discusses how these methods perform for specific equations related to MHD. For example, what happens if I solve the induction equation with an implicit Euler and centred differences? What changes if I use an upwind stencil instead? The book by Durran does a really great job answering such questions for GFD - I was hoping something similar might be around for MHD, too.
PS: I found the following question which is interesting (I will try out the codes linked there), but does not provide an answer for a good book to understand what is happening in the codes linked there.