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I am looking for a good, relatively modern, review paper/book on Finite Difference Methods for PDEs with a theoretical emphasis in mind. By theoretical emphasis I mean that I care about theorems (i.e. with proofs) of convergence (and rate of convergence, if available) to an actual solution. A non-modern (late 1950s) example of the sort of review I'm looking for is O. Ladyzenskaja's "The Method of Finite Differences in the theory of partial differential equations".

Any help finding such papers/books is very well appreciated. More broadly, modern reviews on other discrete schemes for solving PDEs are also welcome, provided the reviews are theoretical in nature (in the sense defined above).

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It is hard to recommend a book or paper without knowing your level of mathematical maturity, and what type of PDE you are interested in. If you are looking for a general introductory advanced undergraduate/graduate text, I would recommend

"Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems" by Randall J. LeVeque

It is a very practical book, but he does take the time to prove convergence with rates at least for some linear PDE.

The situation can become much more difficult for nonlinear PDE, and the results are scattered through research articles and monographs. A good general survey article covering a broad range of numerical methods (not just finite difference) is

"A review of numerical methods for nonlinear partial differential equations" by Eitan Tadmor

You can download the paper here: http://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01379-4/S0273-0979-2012-01379-4.pdf

For many (if not most) numerical schemes for nonlinear PDE, convergence proofs are simply not available. Sometimes this is because the PDE lacks a rigorous notion of solution and well-posedness theory (e.g., mean curvature motion of networks), and in other cases the right ideas for a proof have yet to be discovered (e.g., high order schemes for Hamilton-Jacobi equations, like ENO schemes). It's common for authors of papers to just check consistency of the scheme, and then check that it converges for some test cases.

If there is a particular PDE or class of PDE you are interested in, then perhaps I could refine my recommendation.

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  • $\begingroup$ I am a junior Ph.D student and I've had a lot of practice with the finite difference method, both in practice and in theory. I am familiar with the book by LeVeque, for instance, and it aims a little too low for what I'm looking for. I was not aware of Tadmor's review; that review is more or less the sort of thing I'm looking for, so thanks for sharing it! I am mainly interested in convergence of f.d. schemes for general linear elliptic or parabolic PDEs. O. Ladyzenskaja's review listed in my question is useful in this regard, but potentially outdated w.r.t. best results available. $\endgroup$ – Lentes Oct 17 '16 at 1:14
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    $\begingroup$ For linear elliptic or parabolic in divergence form, I think finite element methods are more commonly used. If you are interested in nonlinear (degenerate) elliptic or parabolic, there is a lot of active research in this area and a good survey is "Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations" Feng, Glowinski, Neilan $\endgroup$ – Jeff Oct 17 '16 at 1:42
  • $\begingroup$ Thanks a lot! I will make sure to take a look at that. $\endgroup$ – Lentes Oct 17 '16 at 2:08

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