David Ketcheson
• Member for 10 years, 1 month
• Last seen this week
• Thuwal, Saudi Arabia

I used to implement everything myself, but lately have begun using libraries much more. I think there are several very important advantages of using a library, beyond just the issue of whether you ...

In the solution of nonlinear hyperbolic PDEs, discontinuities ("shocks") appear even when the initial condition is smooth. In the presence of discontinuities, the notion of solution can only be ...

The equation you're solving does not permit right-going solutions, so there is no such thing as a reflecting boundary condition for this equation. If you consider the characteristics, you'll realize ...

J.M's comment is right: you can find an interpolating polynomial and differentiate it. There are other ways of deriving such formulas; typically, they all lead to solving a van der Monde system for ...

This question may be too broad to have a meaningful answer. Most people who answer will only be familiar with some subset of all the kinds of FD and FE discretizations that may be used. Note that ...

Disclaimer: I sometimes get annoyed when somebody tries to tell me what they think I ought to do rather than answering the question I asked. But I'm going to take a risk and suggest an alternative to ...

van der Houwen's statement is correct, but it is not a statement about all fifth-order Runge-Kutta methods. The "Taylor polynomials" he is referring to are (as you seem to know) just the polynomials ...

There are thousands of papers and hundreds of codes out there using Runge-Kutta methods of fifth order or higher. Note that the most commonly used explicit integrator in MATLAB is ODE45, which ...

Bounds That is still true. In Butcher's book, page 196, it says the following: In a 1985 paper, Butcher showed that you need 11 stages to get order 8, and this is sharp. For order 10, Hairer derived ...

This is a well-framed question and a very useful thing to understand. Korrok is correct to refer you to von Neumann analysis and LeVeque's book. I can add a bit more to that. I'd like to write a ...

There are two main classes of solutions to be discussed in this regard. "Sufficiently" Smooth Solutions In Strang's classical paper it is shown that the Lax equivalence theorem (i.e., the idea that ...

It's quite common in computational fluid dynamics to use implicit schemes similar to what you propose. The ones I know of are based on compact finite difference formulas (not simply on replacing $n$ ...

@Geoff gives a good answer, but I think it's worth providing an alternative perspective. I do everything on Macs -- in OS X, not a Linux VM -- including lots of scientific code development. I mostly ...

Look at the fast multipole method. It is highly scalable and $O(n)$. It allows trading off between precision and cost. Here's an example where it is run at 42 Tflops on a GPU cluster.

The most common method is to reset negative values to some small, positive number. Of course, this is not a mathematically sound solution. A better general approach that may work and is easy, is to ...

Check out HPC University. In particular, the resources section, which includes things like an Introduction to OpenMP Debugging Serial and Parallel Codes Introduction to the Open Science Grid and ...

Good is a relative term, and it will depend on the nature of the problem, the nature of the algorithm, and properties of the hardware involved. The only absolute reference point is ideal scaling (100%...

I wrote a full answer (below the line) before discovering CVXPY, which (like CVX for MATLAB) does all the hard stuff for you and has a very short example almost identical to yours here. You only need ...

This is expected behavior with the Verlet algorithm. It is a symplectic integrator, which means that it will preserve quadratic invariants to within roundoff error -- thus the form planetary orbits ...

I am not an expert in machine learning, but I can outline the considerations that are relevant. The numerical calculations in machine learning are generally linear algebra -- either solving linear ...

Is the shooting method the only general numerical method for solving BVP of nonlinear ODE(s)? No. Most other methods consist of three parts: Discretization. This may be done with finite ...

To get something that looks realistic for planetary orbits, you shouldn't use the forward or backward Euler methods. These will cause your planets to spiral outward or inward. You should use a ...

The boycott of Elsevier should not be mistaken as a push for open access journals. The phrase "open access" does not appear in any of the reasons given for the boycott at http://thecostofknowledge....

The principal numerical difficulty in solving a nonlinear first-order system of hyperbolic PDEs like the Euler equations (for compressible, inviscid flow) is that discontinuities (shock waves) appear ...

Very short answer: for a comprehensive reference, you can't beat Hairer and Wanner volume II. Short answer: Here are some MATLAB scripts to plot the stability region of a linear multistep or Runge-...

This is not an answer to your question. But if a student came to me asking this question, I would ask him or her to read these articles: You and your research (Hamming) How to choose a good ...

The most significant reference I know of is David Stewart's thesis, which is more than 20 years old: High Accuracy Numerical Methods for Ordinary Differential Equations with Discontinuous Right-...

What one usually wants in this situation is to preserve a discrete analog of time symmetry: namely, if the time discretization is applied to solve first forward and then backward in time, the initial ...

The following two methods are given in Functions of Matrices: Theory and Computation by Nicholas Higham, on page 81. These formulas evaluate r(X) = b_0 + \frac{a_1 X}{b_1+\frac{a_2 X}{b_2+\cdots + ...