r"""
Burgers' equation and the Hopf-Cole transformation
==================================================

J. M. Burgers (1948) proposed viscous Burgers' equation as the simplest
model of nonlinear steepening balanced by viscosity. Eberhard Hopf
(1950) and Julian Cole (1951) independently found that
the substitution :math:`u = -2\nu\,\varphi_x/\varphi` turns the
*nonlinear* viscous Burgers' equation :math:`u_t + u u_x = \nu u_{xx}`
into the *linear* heat equation :math:`\varphi_t = \nu\varphi_{xx}`. A
nonlinear PDE with shock-like fronts thus has an exact solution. This
script uses it to watch a sine wave form a viscous shock whose width
shrinks with :math:`\nu`, and to verify the pseudo-spectral Burgers
solver to near machine precision.
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.pde import BurgersEquation1D, burgers_cole_hopf_solution

# %%
# Viscous shocks, exactly
# -----------------------

fig1, ax1 = plt.subplots(figsize=(7, 4))
for nu, color in ((0.3, "tab:blue"), (0.1, "tab:green"), (0.04, "tab:red")):
    x, u = burgers_cole_hopf_solution(np.sin, t=1.5, nu=nu, n=512)
    ax1.plot(x, u, color=color, label=rf"$\nu = {nu}$")
ax1.plot(x, np.sin(x), "k--", lw=1, label="initial data")
ax1.set_xlabel("$x$")
ax1.set_title(r"Hopf-Cole: the front at $x = \pi$ sharpens as $\nu \to 0$")
ax1.legend()
fig1.tight_layout()

# %%
# An exact benchmark for the numerical solver
# -------------------------------------------

nu, t = 0.1, 1.0
for n in (32, 64, 128):
    x, exact = burgers_cole_hopf_solution(np.sin, t=t, nu=nu, n=n)
    numerical = BurgersEquation1D(np.sin, n=n, nu=nu).solve(t, dt=1e-3).final
    print(f"n = {n:3d}: max |pseudo-spectral - Hopf-Cole| = {np.max(np.abs(numerical - exact)):.1e}")

plt.show()