In Stodden, V. 2009. “The Legal Framework for Reproducible Scientific Research.” CiSE. Victoria Stodden recommends publishing the full "research compendium", and lists the following components on p. ...

How can I guarantee that the convergence is uniform "everywhere" on the grid? to confirm the theory results, should not I measure the error at each point and see I have a linear convergence in order ...

Answer to question 1: If $A$ is a circulant matrix, then its eigenvectors are of the form $v_{j,w} = e^{ijhw}$; i.e., they are discrete Fourier modes. So Fourier analysis (or von Neumann analysis, ...

There are existing implementations, though I don't know of any that are publicly available. You could ask Glaser and Rohklin for their implementation of the methods described in this paper.

The essential difference between dense and sparse linear systems is that dense systems express relationships in which each unknown depends on all (or many) of the other unknowns, while sparse systems ...

You could use PyClaw, a parallel extension of Clawpack (note: I am one of PyClaw's main developers). It includes 2D and 3D solvers for the inviscid Euler equations (compressible flow) of an ideal gas....

Wikipedia has an answer here: http://en.wikipedia.org/wiki/Basis_set_(chemistry)#Correlation-consistent_basis_sets Edit: adding introductory text from Wikipedia: Some of the most widely used ...

The other question you have referred to is about the (nonlinear) shallow water equations. Here you are just asking about the linear wave equation, which is quite different. To get a purely right-...

Backward Euler and the implicit trapezoidal rule are both unconditionally stable for this problem. If you're seeing instability then you haven't implemented them correctly.

Generically, yes. Since you have a nonlinear PDE you will end up with a nonlinear algebraic system no matter what spatial discretization you use. With WENO you will have a more strongly nonlinear ...

It depends on exactly what you want to validate. Usually one wishes to confirm convergence of the full discretization, which requires simultaneous refinement in space and time. I would recommend doing ...

I wouldn't say that it is a trick (interpolation is one of the most basic numerical techniques) and I don't see why it would be "dirty" (though it may be inaccurate). I don't think there is a special ...

A new resource for high-order splitting schemes that lists quite a few can be found here: http://www.asc.tuwien.ac.at/~winfried/splitting/

This look like that it is exactly the same as the first order time splitting scheme except the first and last half time step, and the computation is faster with the reduction. Am I missing something ...

If I understand correctly, you are using a centered finite difference in space and the implicit trapezoidal method in time. That scheme is unconditionally absolutely stable, but will generate ...

Your updated question implies that you need to know the current time very precisely -- so precisely that a difference on the order of $10^{-14}$ is unacceptable. If you need to track something to an ...

IMEX methods based on BDF methods have been around for a long time; for a relatively recent discussion take a look at this paper by Ascher, Ruuth, & Spiteri. There are higher-order accurate BDF ...

It sounds like there are two things you might want to "use" them for: To implement a method. Any reference will give you a clear algorithmic description that should make this easy. To check the ...

There are two sources of error, and each of them will be different for your two discretizations. Truncation error, also referred to as discretization error. This results from the fact that you ...

No. Clawpack is designed to solve systems of first-order hyperbolic PDEs on structured (logically quadrilateral or hexahedral) meshes. It is possible to incorporate parabolic terms, but the real ...

Since you have a scalar problem in one space dimension, it shouldn't be too difficult to come up with a semi-analytic solution using the method of characteristics. Here's something to get you started....

You have discretized an advection equation using a forward difference in time and centered differences in space. You have correctly deduced that this is an unstable discretization; in fact it is ...

There are two possibilities: You have a bug. You are seeing an accumulation of discretization errors. Without more detail, it is impossible for us to tell you which is actually happening in your ...

You could try using CVX, which would allow you to code it in exactly the form you've written it (i.e., with $X$ as a matrix rather than a vector). It probably would be solved using a more general ...
I will just sum up the useful remarks made in the comments. Both approximations $E=\frac{1}{2}\left(\left(\frac{\phi_{n}-\phi_{n-1}}{t_{n}-t_{n-1}}\right)^2+V(\phi_n)\right)$ and $E=\frac{1}{2}\... View answer Accepted answer 2 votes The MATLAB routines starting with 'ode', like ode15i, are for solving initial value problems. If you want to solve a boundary value problem, use bvp4c or bvp5c. View answer 2 votes For some nice examples of where high-precision arithmetic has been useful in mathematics, take a look at the book Mathematics by Experiment by Jonathan Borwein and David Bailey. There's also this ... View answer 2 votes I would sum this up as a coupled system of advection-reaction equations. You should only need a boundary condition in$x$at the boundary where characteristics flow into the domain, which depends on ... View answer Accepted answer 2 votes Yes, you've done something wrong. The appropriate 2-norm of a vector that approximates a grid function is $$\|e\|_2 = \sqrt{dx dx \sum_{i,j} e_{ij}^2}.$$ You are incorrectly using$e_{ij}$instead ... View answer 2 votes The value$x=[0,0,0,0]^T$is an equilibrium solution of your system of ODEs. So MATLAB is computing and plotting the answer -- all zeros. It appears that some entries in your initial vector$x_0\$ ...