tberlok/psecas

A framework for solving numerical eigenvalue problems using pseudo-spectral methods.

psecas's Issues

Add Hermite functions

Improve call to solver

When solving the full eigenvalue problem, the solve method should store all the eigenvalues and eigenvectors in a convenient format.

We need an iterate_solver method which increases the resolution until the first M modes have converged to a specified tolerance.

Right now we are wasting computational power when we call solver.solve() in a loop over different modes.

Convert solve into solve_with_guess() and solve()

Chebyshev rational grid

It seems that this works best with the input N odd. Figure this out and
make it a requirement if that is the case.

Improve check for eigenvalue convergence

See equation 7.19 and sections 7.5-7.6 in Boyd, page 136-138.

Add explanations in examples

All examples need a brief text explaining the setup and a reference to the corresponding paper.

Fourier grid interpolation

There is a bug when the Fourier grid starts at negative values. Should be easy to fix.

Add Laguerre functions

Add Chebyshev grid for semi-infinite domain

The best explanation of these are found on page 369 in Boyd.

Add Legendre grid

See Appendix F.10 in Boyd.

Tests of differentiation and interpolation using this grid should be included.

Correct digits in omega

omega is currently stored as double precision along with a tol and atol estimate.
Would be cool to also show how many of the digits are the same as in the previous iteration.

Clean up plottting.py

This file should only contain plotting functionality!

Solve equation 8.30 in Pringle and King

Improve iterative solver

The iterative solver starts using a guess when the difference to the previous iteration is less than a specified guess tolerance. The default for this value should be set to a lower value (it's currently 10 percent).

We should make sure that the difference is always decreasing when we are using guesses. Otherwise we are not converging and we should use the full eigenvalue solver.

See also sections 7.7 and 7.8 in Boyd!

Add sinc grid

It might be interesting to add the sinc grid.

According to Boyd, boundary conditions that the variables goes to zero at \pm \infty is
automatically satisfied when using the sinc grid. This has the advantage that the
evp is a standard evp, not a generalized evp.

See Appendix F.7 in Boyd.

Generalize the boundary condition options

The boundary condition options are currently to fix variables to be zero at zmin and zmax
or to not do anything.

This should be changed such that the boundary condition at zmin and zmax are specified
independently. Furthermore, an option for specifying that the derivative of the variable is zero
should be implemented.

Improve Laguerre and Hermite grids

We currently use dmsuite for these grids. This suite however fails for N>100 or so.

Section 7.5 in Polynomial Approximation of Differential Equations by Daniele Funaro describes how to evaluate the derivative matrices for high N.

Solve equation 9.34 in Pringle and King

Use sympy to test differentation

Use sympy to make the differentation simpler and generalize to higher order derivatives.

Coordinate mappings

Figure out a neat way to enable mappings such as the arcsinh mapping in Boyd.

Equation substitutions

For long equations it would be useful to be able to define shorthands which Psecas then substitutes into the equations before evaluation. This should be easy to implement.

See also #27 which might be related.

Hermite functions

Maybe consider improving the Hermite function basis.

See page 66 in Boyd where he writes:

The simplest definition of Hermite functions (integer coef- ficients) unfortunately leads to normalization factors which grow wildly with n, so it is quite helpful to use orthonormalized Hermite functions. (“Wildly” is not an exaggeration; I have sometimes had overflow problems with unnormalized Hermite functions!) From a theoretical standpoint, it does not matter in the least whether the basis is orthonormal or not so long as one is consistent.

Fix docstring chebyshev_roots

Docstring of chebyshev_roots.py class is the same as chebyshev_extrema.py.
Note: Gauss is misspelt.

Optimize implementation of Chebyshev grids

We currently have some very old code with for loops. It would be good to write this as a matrix operation.

Alternatively we could consider using dmsuite which is probably better, in any case.

Solve equation 5.54 in Pringle and King

Long term storage of data

The current way of storing data, while very convenient, is not optimal for long term storage. Do something smarter, see e.g. here.

mpi io

Create a new log file instead of appending to the existing freja.log

Write the finished statement after all processors have finished. Right now it is written out already when rank 0 is finished.

Make sure that grid.zg[0] < grid.zg[-1]

Some of the grids are reversed. This does not matter for differentation, plotting and so on
but it can lead to issues when we implement boundary conditions!

Calculate polynomial representation of solutions

We currently have an interpolation method for each grid. It would be good to also have a method that returns the coefficients of the Nth degree polynomial representing the data.