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Results tagged with na.numerical-analysis
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user 20507
Numerical algorithms for problems in analysis and algebra, scientific computation
1
vote
FEM based solution to parabolic problem
I know you asked about FE methods, but even simple finite difference methods can deal with this singularity reasonably well. Here is a solution of a 1D problem computed with a 3-point centered differ …
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5
votes
Accepted
Euler method (and others) for unbounded intervals
Regarding 1 and 2:
Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in provably (pointwise) convergent schemes. At least for traditional method …
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1
vote
Accepted
What is the minimum number of stages $s$ required for a Runge-Kutta type numerical method of...
For implicit methods, you can achieve order $2s$ with $s$ stages. Note that this result is the same if one considers the simpler problem of numerical integration (quadrature).
Update as of 2024: a 16 …
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1
vote
Finite difference for a highly nonlinear equation - The wind within the forest
As noted already in the comments, your boundary conditions seem off. Note that generically for a second-order BVP one expects to impose only two boundary conditions; you have 4.
Once you’re sure you …
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1
vote
Accepted
Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis
A bound of the kind you're looking for is not possible, and it doesn't even matter what interpolation points you use (much less which particular polynomial basis you choose to work with, since that ha …
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4
votes
Books and resources on PDEs that use Mathematica and Matlab
For numerical analysis, I can recommend:
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems by R. J. LeVeque.
Spectral methods with M …
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4
votes
Accepted
Benchmark Systems for ODE Solvers - Reference Request
There is a whole subfield of applied mathematics devoted to developing ODE solvers and understanding their properties. Consequently, there are thousands of relevant papers, and not much more can be s …
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2
votes
Accepted
Lower bounds for finite difference formulas
I believe you are asking the following:
What is the minimum number of evaluations of $f$ required to approximate $f^{(k)}$ to order of accuracy $p$?
In fact, this is a homework problem I often g …
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2
votes
Accepted
root solving without analytic derivative
Obviously, within the realm of piecewise-smooth functions one can find examples where any derivative-based approach fails. I believe you're looking for the term "derivative free optimization".
He …
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2
votes
Interpolation by rational functions reference
For a recent reference that includes efficient computational techniques developed in the last few years, see Chapters 26-27 of L. N. Trefethen's Approximation Theory and Approximation Practice. You c …
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2
votes
Frozen coefficient method (von Neumann stability analysis)
I believe the intended reference regarding parabolic PDEs is:
Fritz, John. On integration of parabolic equations by difference methods: I. Linear and quasi-linear equations for the infinite inter …
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2
votes
Solving a simple Schrödinger equation with Fast Fourier Transforms
If I understand correctly, what you're doing amounts to:
Starting with initial data $\psi(x,0)$ that is represented as a truncated Fourier series.
Computing the exact (up to roundoff errors) time ev …
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5
votes
Difference stencils approximating Laplacian
You have assumed an equidistant grid in one dimension, but the answer below can be formulated (and is true) for general grids in any number of dimensions. You also haven't specified the boundary cond …
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9
votes
Current Research in Numeric Mathematics
Most of the problems to which mathematics is applied nowadays cannot be solved symbolically; think of flow in an oil reservoir or air flow past a vehicle. These problems are modelled by complicated s …
7
votes
Computing Gauss Legendre quadrature for large $N$
To add to Fredrik Johansson's answer: A nice history of algorithms for computing Gauss quadrature rules can be found in this SIAM News article by Alex Townsend. Therein, it is stated that the "final …
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