polyopt

Given a spectrum (typically corresponding to a spatial semi-discretization of a PDE), finds an optimal stability polynomial. The polynomial coefficients can then be used as input to RK-coeff-opt to find a corresponding Runge-Kutta method.

This is the implementation of the algorithm described in [ketcheson-ahmadia]. The code was written by Aron Ahmadia and David Ketcheson.

opt_poly_bisect

function [h,poly_coeff] = opt_poly_bisect(lam,s,p,basis,varargin)

Finds an optimally stable polynomial of degree s and order p for the spectrum lam in the interval (h_min,h_max) to precision eps.

Optional arguments:

lam_func:
A function used to generate the appropriate spectrum at each bisection step, instead of using a fixed (scaled) spectrum. Used for instance to find the longest rectangle of a fixed height (see Figure 10 of the CAMCoS paper).

Examples:

  • To find negative real axis inclusion:

    lam = spectrum('realaxis',500);
    s = 10; p = 2;
    [h,poly_coeff] = opt_poly_bisect(lam,s,p,'chebyshev')
    
  • To reproduce figure 10 of [ketcheson-ahmadia]

    lam_func = @(kappa) spectrum('rectangle',100,kappa,10)
    [h,poly_coeff] = opt_poly_bisect(lam,20,1,'chebyshev','lam_func',lam_func)
    plotstabreg_func(poly_coeff,[1])

spectrum

function lamda = spectrum(name,N,kappa,beta)

Return N discretely sampled values from certain sets in the complex plane.

Acceptable values for name:
  • ‘realaxis’: \([-1,0]\)
  • ‘imagaxis’: \([-i,i]\)
  • ‘disk’: \({z : |z+1|=1}\)
  • ‘rectangle’: \({x+iy : -\beta \le y \le \beta, -\kappa \le x \le 0}\)
  • ‘Niegemann-ellipse’ and ‘Niegemann-circle’: See Niegemann 2011
  • ‘gap’: Spectrum with a gap; see Ketcheson & Ahmadia 2012

kappa and beta are used only if name == ‘rectangle’

test_polyopt

function test_suite = test_polyopt

A set of verification tests for the polyopt suite.

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