serendipyty.seismic.utils.fd¶

Created on Wed Dec 7 11:35:44 2016

@author: Filippo Broggini (ETH Zürich) - filippo.broggini@erdw.ethz.ch

Functions

serendipyty.seismic.utils.fd.stability(vmax, dx, dt)[source]¶

Compute maximum dt for a stable simulation.

Parameters:	vmax: float Maximum velocity of the medium dx: float Grid discretization. dt: float Temporal discretization.
Returns:	dt_stable: float Maximum temporal discretization.

serendipyty.seismic.utils.fd.dispersion(vmin, dx, fc, coeff=2.0)[source]¶

Compute maximum dt for a stable simulation.

Parameters:	vmin: float Minimum velocity of the medium dx: float Grid discretization. fc: float Central (peak) frequency of the source wavelet. coeff: float Coefficient to compute the maximum frequency of the wavelet: fmax = coeff * fc.
Returns:	dt_stable: float Maximum temporal discretization.

serendipyty.seismic.utils.fd.fdtaylorcoeff(k, xbar, x)[source]¶

Compute coefficients for finite difference approximation.

Compute coefficients for finite difference approximation for the derivative of order k at xbar based on grid values at points in x.

Notes

This function returns a row vector c of dimension 1 by n, where n=length(x), containing coefficients to approximate \(u^{(k)}(xbar)\), the k’th derivative of u evaluated at xbar, based on n values of u at x(1), x(2), … x(n). If U is a column vector containing u(x) at these n points, then c*U will give the approximation to \(u^{(k)}(xbar)\). Note for k=0 this can be used to evaluate the interpolating polynomial itself. Requires length(x) > k. Usually the elements x(i) are monotonically increasing and x(1) <= xbar <= x(n), but neither condition is required. The x values need not be equally spaced but must be distinct. This program should give the same results as fdcoeffV.m, but for large values of n is much more stable numerically. Based on the program “weights” in [R911].

Note: Forberg’s algorithm can be used to simultaneously compute the coefficients for derivatives of order 0, 1, …, m where m <= n-1. This gives a coefficient matrix C(1:n,1:m) whose k’th column gives the coefficients for the k’th derivative. In this version we set m=k and only compute the coefficients for derivatives of order up to order k, and then return only the k’th column of the resulting C matrix (converted to a row vector). This routine is then compatible with fdcoeffV. It can be easily modified to return the whole array if desired.

From http://www.amath.washington.edu/~rjl/fdmbook/ (2007).

References

[R911]

(1, 2) B. Fornberg, “Calculation of weights in finite difference formulas”, SIAM Review 40 (1998), pp. 685-691.