Plotting Approximate Stability Regions

Copyright (C) 2020 Andreas Kloeckner

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In [1]:
import numpy as np

import matplotlib.pyplot as pt



from cmath import exp, pi
In [2]:
def approximate_stability_region_1d(step_function, make_k, prec=1e-5):

    def is_stable(k):

        y = 1

        for i in range(20):

            if abs(y) > 2:

                return False

            y = step_function(y, i, 1, lambda t, y: k*y)

        return True

    

    def refine(stable, unstable):

        assert is_stable(make_k(stable))

        assert not is_stable(make_k(unstable))

        while abs(stable-unstable) > prec:

            mid = (stable+unstable)/2

            if is_stable(make_k(mid)):

                stable = mid

            else:

                unstable = mid

        else:

            return stable



    mag = 1

    if is_stable(make_k(mag)):

        mag *= 2

        while is_stable(make_k(mag)):

            mag *= 2



            if mag > 2**8:

                return mag

        return refine(mag/2, mag)

    else:

        mag /= 2

        while not is_stable(make_k(mag)):

            mag /= 2



            if mag < prec:

                return mag

        return refine(mag, mag*2)
In [3]:
def plot_stability_region(center, stepper):

    def make_k(mag):

        return center+mag*exp(1j*angle)



    stab_boundary = []

    for angle in np.linspace(0, 2*np.pi, 100):

        stable_mag = approximate_stability_region_1d(stepper, make_k)

        stab_boundary.append(make_k(stable_mag))

        

    stab_boundary = np.array(stab_boundary)

    pt.grid()

    pt.axis("equal")

    pt.plot(stab_boundary.real, stab_boundary.imag)
In [4]:
def fw_euler_step(y, t, h, f):

    return y + h * f(t, y)



plot_stability_region(-1, fw_euler_step)
In [5]:
def heun_step(y, t, h, f):

    yp1_fw_euler =  y + h * f(t, y)

    return y + 0.5*h*(f(t, y) + f(t+h, yp1_fw_euler))



plot_stability_region(-1, heun_step)
In [6]:
def rk4_step(y, t, h, f):

    k1 = f(t, y)

    k2 = f(t+h/2, y + h/2*k1)

    k3 = f(t+h/2, y + h/2*k2)

    k4 = f(t+h, y + h*k3)

    return y + h/6*(k1 + 2*k2 + 2*k3 + k4)



plot_stability_region(-1, rk4_step)
In [ ]: