README

Python 1-D Demo for Adaptive Wavelet Collocation Method (AWCM) (Vasilyev & Bowman, 2000)

Written In Matlab:    Oleg V. Vasilyev    |   14 October, 2018
Converted to Python:  Ari Nejadmalayeri   |   29 November, 2023

The distribution contains the following Python files, essential for understanding of the basics of the AWCM method:

C_range_corr.py         - index correction for update stege of wavelet transform on an interval
D_range_corr.py         - index correction for predict stege of wavelet transform on an interval
G_range_corr.py         - index correction for derivative calculations on an interval
adapt_grid.py           - interal procedure for grid adaptation used to solve PDEs (high level call for convenience)
add_adjacent.py         - adds adjacent wavelets to the mask of significan coefficients
add_ghost.py            - creates a mask of ghost points and index of resolution levels for derivative calculation
adjust_lev.py           - adjusts level of resolution based on analysis of significant wavelet coefficients 
calc_derivative.py      - calculates derivative with on the fly calculation of the mask of ghost points and index of resolution levels 
calc_derivative_mask.py - same as calc_derivative but with precalculation of the mask of ghost points and index of resolution levels 
d2f_pt.py               - finite difference calculation of second derivatig on arbitrary stencil
d2wgh.py                - weights for finite difference calculation of the second derivatig on arbitrary stencil
df_pt.py                - finite difference calculation of first derivatig on arbitrary stencil
diff_wlt.py             - calciulates derivative using global definitions (high level call for convenience)
dwgh.py                 - weights for finite difference calculation of the first derivatig on arbitrary stencil
prd_corr.py             - periodic index correction that wraps the points around
reconstruction_check.py - check if all the supporting wavelets required for adaptive wavelet transform are present and does the correction if necessary.
significant.py          - creates the mask of significant points
test_PDE.py             - script to test soluion of PDE (example of Burger's equation)
test_adapt_grid.py      - script to test grid adaptation for both periodic and interval cases
test_calc_derivative.py - script to test derivative calculation for both periodic and interval cases
wgh.py                  - weights for predict and update stages of wavelet transform for arbitrary stencil
wtf.py                  - wavelet transform 1-d second generation wavelet transform on adaptive grid

All demo files are written for clarity of understanding of the algorithm without any consideration for efficiency.

References:
Vasilyev, O.V. and Bowman, C., “Second Generation Wavelet Collocation Method for the Solution of Partial Differential Equations,” Journal of Computational Physics, 165, pp. 630–693, 2000.