110 lines
2.6 KiB
Python
Executable File
110 lines
2.6 KiB
Python
Executable File
#!/usr/bin/env python3
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import numpy as np
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def pdeHyperbolic(a, x, t, f, g, dtype=np.float64):
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""" Solve a Hyperbolic Partial Differential using finite differences. """
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m = len(x) # Amount of objects to track
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n = len(t) # Length of Time Vector
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# Determine stepsizes
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h = x[0] - x[1]
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k = t[0] - t[1]
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λ_sq = (a*k/h)**2
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# Create array to hold the solution
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w = np.zeros((n,m), dtype=dtype)
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# Create finite difference matrix
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A = np.diag(m*[2*(1 - λ_sq)], k=0) + np.diag((m-1)*[λ_sq], k=-1) + np.diag((m-1)*[λ_sq], k=1)
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# Initialise first two timesteps
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w[0] = f(x, t[0])
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w[1] = A@w[0]/2 + k*g(x, t[0])
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# Calculate for following timesteps
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for j in range(2, n-1):
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w[j] = A@w[j-1] - w[j-2]
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return w
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def test_pdeHyperbolic_case1(x_steps=1e2, t_steps=1e2, max_x=1, max_t=1):
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a = 1 # from the Schroedinger Equation
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# Setup spatial and time grids
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x = np.linspace(0, max_x, x_steps)
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t = np.linspace(0, max_t, t_steps)
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# Boundary conditions
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def f(x,t):
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return np.sin(2*np.pi*x)
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def g(x,t):
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return 2*np.pi*np.sin(2*np.pi*x)
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# Solve it
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sol = pdeHyperbolic(a, x, t, f, g)
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# Plot it with the exact solution
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exact_f = lambda x,t: np.sin(2*np.pi*x)*(np.cos(2*np.pi*t) + np.sin(2*np.pi*t))
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plot_animation(x, sol, func=exact_f)
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def test_pdeHyperbolic_case2(x_steps=2e2, t_steps=4e2, max_x=1, max_t=1):
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a = 1 # from the Schroedinger Equation
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# Setup spatial and time grids
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x = np.linspace(0, max_x, x_steps)
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t = np.linspace(0, max_t, t_steps)
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# Boundary conditions
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def f(x,t):
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return 2*(x < 0.5) -1
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def g(x,t):
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return 0
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# Solve it
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sol = pdeHyperbolic(a, x, t, f, g)
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plot_animation(x, sol)
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def plot_animation(x, sol, func=None, interval = 10):
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from matplotlib import pyplot
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from matplotlib import animation as anim
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fig, _ = pyplot.subplots()
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def animate(i):
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pyplot.clf()
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pyplot.grid()
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pyplot.ylim(-1.5, 1.5)
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pyplot.title('t = {}/{}'.format(i, len(sol)))
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#if func is not None:
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# pyplot.plot(x, func(x, sol[i]), label='exact')
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pyplot.plot(x, sol[i,:], label="iter")
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frames = np.arange(0, len(sol))
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try:
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myAnim = anim.FuncAnimation(fig, animate, frames, interval = interval )
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pyplot.legend()
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pyplot.show()
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except AttributeError:
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# This final error is fugly
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pass
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if __name__ == "__main__":
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test_pdeHyperbolic_case1()
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#test_pdeHyperbolic_case2()
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