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For ODEs I have these books:

I'm looking for references of similar level on the numerical integrations of PDEs. I saw:

All books are edited by Springer, but that's just because I had good experiences in the past with Springer's books for numerical analysis. Feel free to suggest books from other editors if you believe they're a better choice.

Also, at the suggestion of a user, I'll add a few details about the applications I'm interested in. Actually, I don't have a specific application in mind 🙂. I was looking for a general reference, partly because of intellectual curiosity, and partly because in my work there are plenty of opportunities to model many different problems with various PDEs, so there isn't a single one I'm interested in. However, given that for the most complex cases, such as for example turbulent compressible Navier-Stokes, one wouldn't write a code from scratch, but rather use an existing commercial one, I would be mostly interested in the following PDEs:

  • classic linear ones: Laplace, Poisson, Fourier (diffusion or heat equation), D'Alembert (linear waves equation), Helmoltz equation
  • a few "simple" nonlinear ones: Burgers, Buckley–Leverett, diffusion-reaction
  • maybe a couple "not-so-simple" nonlinear ones, such as shallow waters or Euler's, maybe just in 1D

It's OK if the book doesn't cover all of these, but it should at least cover the linear ones and one of the nonlinear ones.

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  • $\begingroup$ Could you add the references of the books instead of just the links? That way we can see then without the bed to go to other site. Also, links might change in the future, but bibliographic information don't. $\endgroup$ – nicoguaro May 26 at 14:11
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    $\begingroup$ What is your application? That could help us recommend some methods over others. $\endgroup$ – nicoguaro May 26 at 14:12
  • $\begingroup$ @nicoguaro I don't have a single specific application in mind. In my work, there are a lot of opportunities to apply numerical integration of PDEs. From heat equation to eikonal equation, to Navier-Stokes equations...however, since for the most complex equations such as turbulent compressible NS one usually resorts to commercial code, let's consider either classic linear PDEs (Laplace, Poisson, Fourier, D'Alembert, Helmoltz, etc.) or simple nonlinear ones (Burgers, Buckley–Leverett , etc.). $\endgroup$ – DeltaIV May 26 at 21:06
  • $\begingroup$ @nicoguaro A link to the book allows people to quickly get an idea of the book's contents. I understand, however, that people reading my question may not be happy about having to click on each link to understand what I'm talking about. For this reason, I added bibliographic information to the links. Thus, people can choose to visit the site, if they like to know more about he books, or to just read the bibliographic information, if they'd rather not. $\endgroup$ – DeltaIV May 26 at 21:18
  • $\begingroup$ @nicoguaro is there anything else you need to provide an answer? $\endgroup$ – DeltaIV May 30 at 17:26
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I don't know of a single book that one would like to have as a unique reference in the topic. Although, I think that the following reference is good enough:

Following there is a (non-exhaustive) list for different methods/topics.

I would start with the finite difference method. For that method I suggest the following reference:

A simple introduction to finite element methods is the following book by the same author:

For finite volumes and hyperbolic problem the following book commonly used as a reference:

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  • $\begingroup$ So, first of all thank you for the answer. I didn't know the first book you referred to, and the second one is even free! However, I see you mention Randy LeVeque's books, which I forgot about. So I had a look at his page and I found this gem: faculty.washington.edu/rjl/fdmbook What do you think about this? It seems similar to the second one you suggested me, but maybe even better. What's your opinion? If you could read only one of them, which one would you choose? $\endgroup$ – DeltaIV May 31 at 8:22
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    $\begingroup$ @DeltaIV, that is a really good book. It is more analysis oriented than the one I suggested for FDM. $\endgroup$ – nicoguaro May 31 at 14:42
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    $\begingroup$ I would highly recommend Randy's FDM book if you are getting started with numerical PDEs. It introduces most of the issues one finds with numerical PDEs, keeps it simple with finite differences, but most of it applies with the other methods such as finite elements and finite volumes. FWIW I was his Ph.D. student and teach from this book but have been doing so for years and still like it. $\endgroup$ – Kyle Mandli Jun 1 at 1:09

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