David Ketcheson
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What material should I include with a journal article (or post online) in order to make my computational research reproducible?
3 votes

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. ...

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how to measure the error of a finite difference method
3 votes

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 ...

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eigenvalue analysis vs fourier analysis for stability and their equivalence
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3 votes

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, ...

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What algorithm for solving a set of stiff ODEs would be easiest to port to high precision floating point arithmetic?
3 votes

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.

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Where do dense matrices occur?
3 votes

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 ...

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Is there a good, easy-to-use, high quality open source CFD solver out there?
3 votes

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....

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What does it mean for a basis set to be "correlation consistent" ?
3 votes

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 ...

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A simple wave for the linear shallow water equations
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2 votes

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-...

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Stability Criteria for Numerical Solution of Windkessel Ordinary Differential Equation
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2 votes

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.

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If I discretize a PDE in space with WENO and in time with an implicit method, do I need to solve a nonlinear algebraic system at each time step?
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2 votes

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 ...

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Converge rate analysis: issue with time convergence
2 votes

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 ...

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citations for numerical lookup table interpolation of P/ODE(s) RHS
2 votes

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 ...

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Are there operator splitting approaches for multiphysics PDEs that achieve high order convergence?
2 votes

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/

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Strang splitting
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2 votes

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 ...

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Numerical solution of burgers equation with finite volume method and crank-nicolson
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2 votes

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 ...

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Accurate computation of the current time in time integrator
2 votes

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 ...

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BDF methods for implicit-explicit method
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2 votes

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 ...

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Good tutorials on how to use Butcher tables?
2 votes

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 ...

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Why do structured and unstructured discretizations give different errors?
2 votes

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 ...

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Application of CLAWPACK to Richards' equation
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2 votes

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 ...

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How can I solve this 1D nonlinear, variable-coefficient hyperbolic PDE?
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2 votes

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....

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Finite differences scheme for 2D advection equation
2 votes

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 ...

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scipy odeint: sum of conservative ode equations does not remain zero as it is being solved -- is this normal?
2 votes

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 ...

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Linear programming with matrix constraints
2 votes

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 ...

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Numerical scheme with energy conservation?
2 votes

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}\...

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How do I solve a boundary value ODE in MATLAB?
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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.

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Relevance of fixed-point and arbitrary precision computations
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 ...

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mathematical statement of "open" boundary condition
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 ...

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rate of convergence for the second order accurate method on two dimensional grid
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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 ...

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problem in output solving two coupled second order differential equations in matlab
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 $x_0$ ...

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