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I'm currently working on my MS in CS and have developed an interest in astrophysics. Luckily one of my professors is a astrophysicist and is currently doing research through computational physics and he agreed to help me get started.

I'm currently reading through Tao Pang's "An Introduction to Computational Physics" (TOC), but the book's lack of detail on the mathematical methods used leaves me pretty lost at times.

My question is, are there any books that cover the topics in a more in-depth fashion or any books that I could use as a reference for the mathematical methods?

EDIT: The best way to describe my mathematical background is that I've studied calculus, linear algebra and differential equations, but remember very little.

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  • $\begingroup$ You haven't told us much about your mathematical background so it's hard to tell whether the difficulty that you're having is related to lack of mathematical background, but it's quite common for students in CS to have weak backgrounds in calculus, linear algebra, and differential equations. If those kinds of issues are where the book is not providing enough detail, then the best approach is probably to go back and study those subjects first. It's a different issue if your concern is that the author isn't providing rigorious analysis to prove that the methods work. $\endgroup$ – Brian Borchers Oct 20 '13 at 19:35
  • $\begingroup$ An example of a particular point in this book that hasn't made sense to you might help us to find a more accessbile reference for you. $\endgroup$ – Brian Borchers Oct 20 '13 at 19:37
  • $\begingroup$ The best way to describe my mathematical background is that I've studied calculus, linear algebra and differential equations, but remember very little. An example would be the La Grange interpolation, I understand what is does, I just can't follow how the formulas are derived. This applies to most of the topics, I understand the gist of the method, but I can't follow how we come to the conclusion. $\endgroup$ – inzombiak Oct 20 '13 at 19:40
  • $\begingroup$ Chances are pretty good that your problems relate to the mathematics that you've forgotten. $\endgroup$ – Brian Borchers Oct 20 '13 at 19:47
  • $\begingroup$ In the case, let me revise my question. Can you suggest any books or resources I can use to refresh my memory? Khan Academy has videos on all 3 of the topics you mentioned, but no real exercises for cementing them into my brain. And is Tao Pang's book worth sticking to, or is there another you would suggest? Even if it is the same level of complexity. $\endgroup$ – inzombiak Oct 20 '13 at 19:55
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From skimming the table of contents to the book you listed, I'd say that computational books of that type for physics (or in my case, engineering, since that is my background) tend to sacrifice depth and quality of explanation for breadth. The best book of this ilk I can think of is probably Strang's Computational Science and Engineering, because he's a great writer of mathematics. Many "numerical methods" classes geared towards scientists and engineers tend to teach just enough math in whatever topics they feel the intended audience needs to get by, and cobble together a bunch of notes. (Or they punt, and teach out of Burden and Faires, or worse, rely on Numerical Recipes.)

To get better explanations and depth, you'll probably need to get access to more specialized textbooks, such as:

A list could be created for optimization (Nocedal's textbook is a good start), finite volume methods (LeVeque's books), finite difference methods (again, LeVeque has another good book, or the book by Strikwerda), finite element methods (not an expert here), and so on. The tradeoff is depth (and quality of explanation) for time. I agree with Pedro that you'll probably be able to make the largest contributions right away in the algorithms and data structures used to implement physics. If you really want to dig into the math, though, I'd pick one of the specialty books that you think will be most useful to you and start reading it.

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Short answer: No, at least not that I know of.

I'm a Computer Scientist who, for the past year and a half, has been working with Astrophysicists on writing faster simulation codes. I spent quite a bit of time looking for books on the topic to provide some sort of overview, and found nothing convincing.

Most of what I now know about Astrophysics, I learnt from asking Astrophysicists directly. There are a number of good review papers on specific topics, e.g. Daniel Price's papers on SPH, e.g. here, or Walter Dehnen's papers on N-body solvers, e.g. here, but I still rely on my colleagues in that field for the details.

What I think you should keep in mind, though, is that most of the interesting computational problems don't involve that much physics or even maths, but algorithms and data structures. Keep in mind that most physics codes (or books on physics codes) were written not by Computer Scientists, but by Physicists. That doesn't necessarily mean they are bad, but given the choice, would you rather buy a physics book written by a Computer Scientist, or by a Physicist?

As a Computer Scientist, you have an edge with regards to algorithms and data structures, and there are loads of low-hanging fruit there. In most of my own work in this area, I have not had to tweak the physics of any problem at all, just the algorithms implementing them. Simply getting the algorithms right can already lead to massive speedups.

In summary: If you have close collaborators who are Astrophysicists, rely on them to get the physics right, but keep in mind that you will probably be able to make the largest contributions in the algorithms and data structures used to implement said physics.

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You might look at James Nearing's book. It's available Here. I don't think it's very academic, but I learned a lot from it. One reviewer said about it: Unlike the usual dry and formal textbook, it reads like a friendly uncle explaining things in plain English, trying to cut through the red tape and tell you how to actually think about this stuff. The graphic about the Taylor's series of a sine function is actually quite moving. For free, what could go wrong?

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You might try: Bertil Gustafsson: Fundamentals of Scientific Computing to get an overview, and for a more encyclopaedic treatment of the mathematical methods Gander, Gander, Kwok: Scientific Computing.

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