For ODEs I have these books:
Griffiths, David, Higham, Desmond J., Numerical Methods for Ordinary Differential Equations, 2010
Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Numerical Mathematics, 2006
I'm looking for references of similar level on the numerical integrations of PDEs. I saw:
Evans, G., Blackledge, J., Yardley,P., Numerical Methods for Partial Differential Equations, 2000
Griffiths, David F., Dold, John W., Silvester, David J., Essential Partial Differential Equations, 2015
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.