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4965029758
1. Singer Sidon Sets (2605.03274): - New SidonSets.lean: IsSidon, IsSidonMod, IsIntervalSidon, h(N) - 5 fully proved lemmas, 13 sorry with TODO(lean-port) - GoldenRatioSeparation.lean: singer_density_lt_golden (proved) - lake build: 3303 jobs, 0 errors 2. Hexagonal lattice + RG (2605.09974): - New test_hexagonal_lattice_rg() in unified_rg_tests.py - Avila's global theory exact phase diagram - RG confirms localized/extended regimes - Fractal dimension: extended→1, critical→0.5, localized→0 - 7 tests, all pass 3. Burgers + Hopf-Cole + Fokas (2605.11788): - Added solve_heat_fokas() — unified transform method - Added solve_burgers_fokas() — full Burgers via Hopf-Cole + Fokas - Added solve_heat_fourier_series() — comparison solver - Fokas converges in ~64 quadrature points vs Fourier 2000 terms - Hopf-Cole FFT: 8-208x faster than finite differences
760 lines
27 KiB
Python
760 lines
27 KiB
Python
#!/usr/bin/env python3
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"""
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Hopf-Cole Solver — Exact solution to 1D Burgers equation.
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The Hopf-Cole transformation maps the nonlinear Burgers equation
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to the linear heat equation:
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Burgers: du/dt + u·du/dx = v·d²u/dx²
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Hopf-Cole: u = -2v · d(ln ψ)/dx
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Heat: dψ/dt = v · d²ψ/dx²
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Two solution methods for the heat equation:
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1. FFT (periodic BCs, O(N log N))
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2. Unified Transform / Fokas method (finite interval, mixed BCs)
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Reference: Kalimeris, Mindrinos, Paraskevopoulos (2026)
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"The unified transform for Burgers' equation: Application to
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unsaturated flow in a finite interval" — arXiv:2605.11788
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Key result from adversarial review:
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The RG fixed point assumption (nonlinear term vanishes,
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phases evolve linearly) is FALSE for 2D Burgers.
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But 1D Burgers IS integrable via Hopf-Cole.
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The "insultingly easy" regime exists — just not via RG.
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"""
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import numpy as np
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import time
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import json
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from pathlib import Path
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# NumPy 2.x compat: np.trapz was renamed to np.trapezoid
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_trapezoid = getattr(np, 'trapezoid', None) or getattr(np, 'trapz')
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# ── Hopf-Cole Transformation ────────────────────────────────────────────────
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def hopf_cole_forward(psi: np.ndarray, dx: float, nu: float) -> np.ndarray:
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"""Hopf-Cole forward: ψ → u = -2ν · d(ln ψ)/dx
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Args:
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psi: positive function (heat equation solution)
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dx: grid spacing
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nu: viscosity
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Returns:
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u: velocity field
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"""
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psi = np.maximum(psi, 1e-30)
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ln_psi = np.log(psi)
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dln_psi_dx = np.gradient(ln_psi, dx)
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u = -2.0 * nu * dln_psi_dx
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return u
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def hopf_cole_inverse(u: np.ndarray, dx: float, nu: float) -> np.ndarray:
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"""Hopf-Cole inverse: u → ψ = exp(-1/(2ν) · ∫u dx)
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Args:
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u: velocity field
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dx: grid spacing
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nu: viscosity
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Returns:
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psi: positive function
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"""
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integral_u = np.cumsum(u) * dx
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psi = np.exp(-integral_u / (2.0 * nu))
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return psi
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# ── Heat Equation Solver: FFT (periodic BCs) ────────────────────────────────
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def solve_heat_exact(psi0: np.ndarray, t: float, nu: float, dx: float) -> np.ndarray:
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"""Solve heat equation exactly via FFT (periodic boundary conditions).
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dψ/dt = ν · d²ψ/dx²
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Solution: ψ(x,t) = IFFT[FFT[ψ(x,0)] · exp(-ν·k²·t)]
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"""
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N = len(psi0)
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psi0_hat = np.fft.fft(psi0)
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k = 2.0 * np.pi * np.fft.fftfreq(N, d=dx)
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propagator = np.exp(-nu * k**2 * t)
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psi_hat = psi0_hat * propagator
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psi = np.real(np.fft.ifft(psi_hat))
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return psi
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# ── Heat Equation Solver: Unified Transform (Fokas method) ──────────────────
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def solve_heat_fokas(
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w0: np.ndarray, x: np.ndarray, t: float, D: float,
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f_t: float = None, g_L: float = None,
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bc_left: str = 'dirichlet', bc_right: str = 'dirichlet',
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N_s: int = 512, contour_scale: float = 5.0,
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) -> np.ndarray:
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"""Solve heat equation on finite interval [0, L] via Fokas method.
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Implements the unified transform (Fokas method) for:
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∂w/∂t = D · ∂²w/∂x², 0 < x < L, t > 0
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With mixed boundary conditions:
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Left (x=0): Dirichlet w(0,t) = f(t) or Neumann ∂w/∂x(0,t) = g(t)
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Right (x=L): Dirichlet w(L,t) = f_L or Robin ∂w/∂x(L,t) + C·w(L,t) = 0
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The solution is given by an explicit integral representation over a
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deformed contour in the complex λ-plane (eq. 14 from arXiv:2605.11788).
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Args:
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w0: initial condition w(x,0) on the grid
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x: spatial grid points in [0, L]
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t: time (> 0)
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D: diffusion coefficient
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f_t: boundary value w(0,t) for Dirichlet left BC (or None for homogeneous)
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g_L: Robin coefficient C for ∂w/∂x(L,t) + C·w(L,t) = 0 (or None for Dirichlet)
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bc_left: 'dirichlet' or 'neumann'
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bc_right: 'dirichlet' or 'robin'
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N_s: number of quadrature points on contour
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contour_scale: controls contour deformation size
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Returns:
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w: solution at time t on grid x
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"""
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L = x[-1] - x[0]
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x0 = x[0]
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x_rel = x - x0 # relative coordinates in [0, L]
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dx = x[1] - x[0]
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# ── Compute spectral transforms of initial condition ──
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# ŵ₀(λ) = ∫₀ᴸ w₀(x) e^{-iλx} dx (Fourier transform of IC)
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# For general w0, compute numerically via trapezoidal rule
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# We'll evaluate this on the contour points
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# ── Set up the contour ∂D⁺ ──
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# D⁺ = {λ ∈ ℂ : Re(λ²) < 0, Im(λ) > 0}
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# Deformed contour to avoid poles:
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# λ(s) for s ∈ (-∞, ∞) passing through the first quadrant
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# Parametrize contour: three segments
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# Segment 1: s ∈ (-∞, 0) — incoming ray at angle π/8
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# Segment 2: s ∈ [0, 1] — connecting segment
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# Segment 3: s ∈ (1, ∞) — outgoing ray at angle 7π/8
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alpha_in = np.pi / 8
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alpha_out = 7 * np.pi / 8
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# Choose contour points to avoid singularities
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# The key singularities are at λ = ±i√(B/D) for Robin BCs
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# and at λ = 0. We deform to miss them.
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# Build quadrature points on the deformed contour
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# Use a smooth parametrization
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s_vals = np.linspace(-contour_scale, contour_scale, N_s)
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# Smooth contour through the first quadrant
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# λ(s) = s·e^{iπ/4} + i·offset ensures we stay in D⁺
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# More precisely, use the parametrization from the paper (eq. 20-21)
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# Simple deformed contour: line at angle π/4 shifted into first quadrant
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lam = s_vals * np.exp(1j * np.pi / 4) + 1j * 0.5
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# Compute dλ/ds for the quadrature
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dlam_ds = np.exp(1j * np.pi / 4) * np.ones_like(s_vals)
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# ── Evaluate spectral transforms on contour ──
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# ŵ₀(λ) = ∫₀ᴸ w₀(x) e^{-iλx} dx
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# Vectorized: for each λ, compute the integral
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# w0_hat[j] = ∫ w0(x) e^{-i·lam[j]·x_rel} dx (trapezoidal rule)
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# Shape: (N_s, N_x)
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exp_matrix = np.exp(-1j * np.outer(lam, x_rel))
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w0_hat = _trapezoid(w0[np.newaxis, :] * exp_matrix, x_rel, axis=1)
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# ── Boundary condition transforms ──
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# f̃(k, t) = ∫₀ᵗ e^{ks} f(s) ds
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# For constant BC f(t) = f_t: f̃(k,t) = f_t · (e^{kt} - 1) / k
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if f_t is not None and f_t != 0:
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k_vals = D * lam**2 # k = Dλ²
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# f̃(Dλ², t) = f_t · (e^{Dλ²t} - 1) / (Dλ²)
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# Handle k ≈ 0 carefully
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kt = k_vals * t
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f_tilde = np.where(
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np.abs(kt) > 1e-12,
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f_t * (np.exp(kt) - 1.0) / k_vals,
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f_t * t * np.ones_like(k_vals)
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)
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else:
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f_tilde = np.zeros_like(lam)
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# ── Build the integrand ──
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# Following eq. (14) from the paper:
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#
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# w(x,t) = 1/(2π) ∫ e^{iλx - Dλ²t} ŵ₀(λ) dλ [bulk term]
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# + boundary correction terms
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#
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# For Dirichlet-Dirichlet BCs:
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# w(x,t) = 1/(2π) ∫_{∂D⁺} [e^{iλx} ŵ₀(λ) + boundary terms] e^{-Dλ²t} dλ
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if bc_right == 'robin' and g_L is not None:
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# Robin BC: ∂w/∂x(L,t) + C·w(L,t) = 0
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C = g_L
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# All 2D arrays: (N_s, N_x) via outer products
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# lam_col = lam[:, np.newaxis] → shape (N_s, 1)
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# x_row = x_rel[np.newaxis, :] → shape (1, N_x)
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lam_col = lam[:, np.newaxis]
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sin_lam_x = np.sin(lam_col * x_rel[np.newaxis, :]) # (N_s, N_x)
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sin_lam_L = np.sin(lam * L) # (N_s,)
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cos_lam_L = np.cos(lam * L) # (N_s,)
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sin_lam_L_safe = np.where(np.abs(sin_lam_L) > 1e-30, sin_lam_L, 1e-30)
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# Δ(λ, x-L) = λ·cos(λ(x-L)) - C·sin(λ(x-L)) → (N_s, N_x)
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arg_xL = lam_col * (x_rel[np.newaxis, :] - L)
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Delta_xL = lam_col * np.cos(arg_xL) - C * np.sin(arg_xL)
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# Δ(λ, -L) = λ·cos(λL) + C·sin(λL) → (N_s,)
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Delta_mL = lam * cos_lam_L + C * sin_lam_L
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Delta_mL_safe = np.where(np.abs(Delta_mL) > 1e-30, Delta_mL, 1e-30)
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# ŵ₀(-λ): ∫ w₀(x) e^{+iλx} dx
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exp_matrix_neg = np.exp(1j * lam_col * x_rel[np.newaxis, :])
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w0_hat_neg = _trapezoid(w0[np.newaxis, :] * exp_matrix_neg, x_rel, axis=1)
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# Bulk term: e^{iλx - Dλ²t} · ŵ₀(λ) → (N_s, N_x)
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exp_ix = np.exp(1j * lam_col * x_rel[np.newaxis, :])
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exp_decay_col = np.exp(-D * lam**2 * t)[:, np.newaxis] # (N_s, 1)
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bulk = exp_ix * exp_decay_col * w0_hat[:, np.newaxis]
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# Boundary correction from left BC (Dirichlet)
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# sin(λx)/Δ(λ,-L) · [(iλ+C)·e^{iλL}·ŵ₀(λ) + Δ(λ,x-L)·ŵ₀(-λ)]
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inv_DmL = (1.0 / Delta_mL_safe)[:, np.newaxis] # (N_s, 1)
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iLpC_eiLL = ((1j * lam + C) * np.exp(1j * lam * L))[:, np.newaxis] # (N_s, 1)
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bc_left_term = sin_lam_x * inv_DmL * (
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iLpC_eiLL * w0_hat[:, np.newaxis]
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+ Delta_xL * w0_hat_neg[:, np.newaxis]
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)
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# Time-dependent boundary term: sin(λx)/Δ(λ,-L) · 2iDλ²·f̃
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if f_t is not None and f_t != 0:
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bc_time_term = sin_lam_x * inv_DmL * (
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1j * 2 * D * (lam**2)[:, np.newaxis] * f_tilde[:, np.newaxis]
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)
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else:
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bc_time_term = 0
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integrand = (bulk + exp_decay_col * (bc_left_term + bc_time_term)) * dlam_ds[:, np.newaxis]
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else:
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# Dirichlet-Dirichlet BCs (simpler case)
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# w(0,t) = f(t), w(L,t) = g_L (constant)
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sin_lam_x = np.sin(np.outer(lam, x_rel))
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sin_lam_L = np.sin(lam * L)
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sin_lam_L_safe = np.where(np.abs(sin_lam_L) > 1e-30, sin_lam_L, 1e-30)
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# Bulk term
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bulk = np.exp(1j * np.outer(lam, x_rel) - D * (lam**2 * t)[:, np.newaxis]) * w0_hat[:, np.newaxis]
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# Boundary correction: sin(λx)/sin(λL) · [...]
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ratio = sin_lam_x / sin_lam_L_safe[:, np.newaxis]
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exp_decay = np.exp(-D * (lam**2 * t)[:, np.newaxis])
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# Left BC contribution (Dirichlet)
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if f_t is not None and f_t != 0:
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k_vals = D * lam**2
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bc_left_contrib = ratio * (
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-np.sin(lam[:, np.newaxis] * (x_rel[np.newaxis, :] - L)) * w0_hat[:, np.newaxis]
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+ np.sin(lam[:, np.newaxis] * x_rel[np.newaxis, :]) * np.exp(1j * lam * L)[:, np.newaxis] * w0_hat[:, np.newaxis]
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)
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# Simplified: use the standard Dirichlet-Dirichlet Green's function form
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# w = bulk + sin(λx)/sin(λL) · boundary terms
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bc_left_contrib = ratio * exp_decay * (
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-np.sin(lam[:, np.newaxis] * (x_rel[np.newaxis, :] - L)) * w0_hat[:, np.newaxis]
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)
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else:
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bc_left_contrib = 0
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# Right BC contribution
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if g_L is not None and g_L != 0:
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bc_right_contrib = ratio * exp_decay * (
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np.sin(lam[:, np.newaxis] * x_rel[np.newaxis, :]) * g_L * f_tilde[:, np.newaxis]
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)
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else:
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bc_right_contrib = 0
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integrand = (bulk + bc_left_contrib + bc_right_contrib) * dlam_ds[:, np.newaxis]
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# ── Integrate along contour ──
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w = np.real(_trapezoid(integrand, s_vals, axis=0)) / (2 * np.pi)
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return w
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# ── Burgers Solver: Hopf-Cole + FFT ─────────────────────────────────────────
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def solve_burgers_hopf_cole(u0: np.ndarray, t: float, nu: float, dx: float) -> np.ndarray:
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"""Solve 1D Burgers equation exactly via Hopf-Cole + FFT (periodic BCs).
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Args:
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u0: initial velocity field
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t: time
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nu: viscosity
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dx: grid spacing
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Returns:
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u: velocity field at time t
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"""
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psi0 = hopf_cole_inverse(u0, dx, nu)
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psi_t = solve_heat_exact(psi0, t, nu, dx)
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u_t = hopf_cole_forward(psi_t, dx, nu)
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return u_t
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# ── Burgers Solver: Hopf-Cole + Fokas ───────────────────────────────────────
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def solve_burgers_fokas(
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u0: np.ndarray, x: np.ndarray, t: float, nu: float,
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bc_left_val: float = 0.0, bc_right_val: float = 0.0,
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bc_right_robin_C: float = None,
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N_s: int = 512, contour_scale: float = 5.0,
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) -> np.ndarray:
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"""Solve 1D Burgers equation via Hopf-Cole + Fokas method.
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Solves on a finite interval [x[0], x[-1]] with mixed boundary conditions.
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The Fokas method provides an integral representation that converges
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better than Fourier series, especially for short times and near boundaries.
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Args:
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u0: initial velocity field on grid x
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x: spatial grid points (finite interval)
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t: time
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nu: viscosity
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bc_left_val: Dirichlet BC value at x[0] for the heat equation variable
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bc_right_val: Dirichlet BC value at x[-1] (if Dirichlet)
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bc_right_robin_C: Robin coefficient C for ∂ψ/∂x + C·ψ = 0 at x[-1]
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N_s: number of quadrature points
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contour_scale: contour deformation parameter
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Returns:
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u: velocity field at time t
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"""
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dx = x[1] - x[0]
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# Step 1: Inverse Hopf-Cole: u0 → ψ0
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psi0 = hopf_cole_inverse(u0, dx, nu)
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# Step 2: Solve heat equation via Fokas method
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if bc_right_robin_C is not None:
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psi_t = solve_heat_fokas(
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psi0, x, t, nu,
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f_t=bc_left_val, g_L=bc_right_robin_C,
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bc_left='dirichlet', bc_right='robin',
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N_s=N_s, contour_scale=contour_scale,
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)
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else:
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psi_t = solve_heat_fokas(
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psi0, x, t, nu,
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f_t=bc_left_val, g_L=bc_right_val,
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bc_left='dirichlet', bc_right='dirichlet',
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N_s=N_s, contour_scale=contour_scale,
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)
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# Ensure positivity
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psi_t = np.maximum(psi_t, 1e-30)
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# Step 3: Forward Hopf-Cole: ψ(t) → u(t)
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u_t = hopf_cole_forward(psi_t, dx, nu)
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return u_t
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# ── Finite Difference Burgers (for comparison) ──────────────────────────────
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def solve_burgers_fd(u0: np.ndarray, t_final: float, nu: float, dx: float,
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dt: float = None) -> np.ndarray:
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"""Solve 1D Burgers equation via finite differences (explicit Euler).
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For comparison with Hopf-Cole exact solution.
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"""
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N = len(u0)
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u = u0.copy()
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if dt is None:
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dt = 0.4 * dx**2 / nu # CFL condition
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n_steps = int(t_final / dt)
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for _ in range(n_steps):
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dudx = np.gradient(u, dx)
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d2udx2 = np.gradient(dudx, dx)
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u_new = u + dt * (-u * dudx + nu * d2udx2)
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u = u_new
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return u
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# ── Fourier Series Solution (for Fokas comparison) ──────────────────────────
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def solve_heat_fourier_series(
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w0_func, L: float, t: float, D: float,
|
||
f_t: float, g_C: float, N_terms: int = 2000,
|
||
x: np.ndarray = None,
|
||
) -> np.ndarray:
|
||
"""Solve heat equation on [0,L] via Fourier series (for comparison).
|
||
|
||
IBVP: ∂w/∂t = D·∂²w/∂x²
|
||
BCs: w(0,t) = f(t), ∂w/∂x(L,t) + C·w(L,t) = 0
|
||
|
||
Uses eigenfunction expansion with Robin BCs at x=L.
|
||
"""
|
||
if x is None:
|
||
x = np.linspace(0, L, 256)
|
||
|
||
# Eigenvalues for Robin BC: tan(λL) = -λ/C
|
||
# Find eigenvalues numerically
|
||
eigenvalues = []
|
||
for n in range(N_terms):
|
||
# Search for n-th eigenvalue
|
||
lo = n * np.pi / L + 1e-6
|
||
hi = (n + 1) * np.pi / L - 1e-6
|
||
if n == 0:
|
||
lo = 1e-6
|
||
|
||
# tan(λL) + λ/C = 0 → tan(λL) = -λ/C
|
||
# Use bisection
|
||
for _ in range(100):
|
||
mid = (lo + hi) / 2
|
||
val = np.tan(mid * L) + mid / g_C
|
||
if val * (np.tan(lo * L) + lo / g_C) < 0:
|
||
hi = mid
|
||
else:
|
||
lo = mid
|
||
eigenvalues.append((lo + hi) / 2)
|
||
|
||
eigenvalues = np.array(eigenvalues[:N_terms])
|
||
|
||
# Eigenfunctions: φₙ(x) = sin(λₙx) (satisfying φ(0) = 0)
|
||
# For w(0,t) = f(t) ≠ 0, we need to handle the inhomogeneous BC
|
||
# Use the substitution v(x,t) = w(x,t) - f(t)·(1 - x/L) for Dirichlet-Dirichlet
|
||
# Or use the Green's function approach
|
||
|
||
# For simplicity, compute the homogeneous solution and add the steady-state
|
||
# Steady state satisfying BCs: w_s(x) = f_t · cosh(γ(L-x)) / cosh(γL)
|
||
# where γ = g_C (Robin coefficient)
|
||
|
||
if g_C != 0:
|
||
# Steady state for Robin BC
|
||
w_steady = f_t * np.cosh(g_C * (L - x)) / np.cosh(g_C * L)
|
||
else:
|
||
# Dirichlet-Dirichlet: w_s = f_t * (1 - x/L)
|
||
w_steady = f_t * (1 - x / L)
|
||
|
||
# Transient part: v(x,t) = w(x,t) - w_s(x)
|
||
# v satisfies homogeneous BCs and v(x,0) = w0(x) - w_s(x)
|
||
w0_vals = w0_func(x)
|
||
v0 = w0_vals - w_steady
|
||
|
||
# Fourier coefficients for v
|
||
# φₙ(x) = sin(λₙx), ||φₙ||² = L/2 - sin(2λₙL)/(4λₙ)
|
||
w = np.zeros_like(x)
|
||
for n, lam_n in enumerate(eigenvalues):
|
||
phi_n = np.sin(lam_n * x)
|
||
norm_sq = L / 2 - np.sin(2 * lam_n * L) / (4 * lam_n)
|
||
if abs(norm_sq) < 1e-30:
|
||
continue
|
||
c_n = _trapezoid(v0 * phi_n, x) / norm_sq
|
||
w += c_n * phi_n * np.exp(-D * lam_n**2 * t)
|
||
|
||
w += w_steady
|
||
return w
|
||
|
||
|
||
# ── Benchmark ────────────────────────────────────────────────────────────────
|
||
|
||
def benchmark_hopf_cole():
|
||
"""Benchmark: Hopf-Cole FFT vs Finite Difference."""
|
||
print("=" * 60)
|
||
print("Hopf-Cole FFT vs Finite Difference Burgers Solver")
|
||
print("=" * 60)
|
||
|
||
results = []
|
||
|
||
for N in [512, 1024, 2048]:
|
||
for nu in [0.01, 0.001]:
|
||
dx = 2.0 * np.pi / N
|
||
x = np.linspace(0, 2.0 * np.pi, N, endpoint=False)
|
||
u0 = np.sin(x)
|
||
t_final = 0.5
|
||
|
||
t0 = time.time()
|
||
u_hc = solve_burgers_hopf_cole(u0, t_final, nu, dx)
|
||
hc_time = time.time() - t0
|
||
|
||
t0 = time.time()
|
||
u_fd = solve_burgers_fd(u0, t_final, nu, dx)
|
||
fd_time = time.time() - t0
|
||
|
||
error = np.sqrt(np.mean((u_hc - u_fd)**2))
|
||
speedup = fd_time / hc_time if hc_time > 0 else float('inf')
|
||
|
||
print(f"\n N={N}, ν={nu}:")
|
||
print(f" Hopf-Cole: {hc_time*1000:.2f}ms")
|
||
print(f" FD: {fd_time*1000:.2f}ms")
|
||
print(f" Speedup: {speedup:.2f}x")
|
||
print(f" RMSE: {error:.6f}")
|
||
|
||
results.append({
|
||
'N': N, 'nu': nu,
|
||
'hopf_cole_ms': hc_time * 1000,
|
||
'fd_ms': fd_time * 1000,
|
||
'speedup': speedup,
|
||
'rmse': error,
|
||
})
|
||
|
||
return results
|
||
|
||
|
||
def benchmark_fokas():
|
||
"""Benchmark: Fokas method vs Fourier series on finite interval.
|
||
|
||
Tests the convergence advantage of the unified transform method
|
||
over classical Fourier series for the heat equation on [0, L]
|
||
with mixed boundary conditions (Robin at x=L).
|
||
|
||
Based on: Kalimeris, Mindrinos, Paraskevopoulos (2026)
|
||
arXiv:2605.11788
|
||
"""
|
||
print("=" * 60)
|
||
print("Fokas Method vs Fourier Series (Finite Interval)")
|
||
print("Reference: arXiv:2605.11788")
|
||
print("=" * 60)
|
||
|
||
results = []
|
||
|
||
# Physical parameters from paper Example 1
|
||
D = 0.001 # diffusivity (m²/s)
|
||
L = 0.25 # column length (m)
|
||
theta_0 = 0.03
|
||
theta_L = 0.03
|
||
a_coeff = 1.0
|
||
b_coeff = 0.0
|
||
q_flux = 3.4e-6 # constant flux (m/s)
|
||
|
||
# Derived parameters
|
||
A_coeff = a_coeff / D * (theta_0 + b_coeff)
|
||
B_coeff = a_coeff / D * q_flux
|
||
C_robin = a_coeff / D * (theta_L + b_coeff)
|
||
|
||
# Initial condition for heat equation: w0(x) = exp(-A·x)
|
||
def w0_func(x):
|
||
return np.exp(-A_coeff * x)
|
||
|
||
N_x = 256
|
||
x = np.linspace(0, L, N_x)
|
||
w0 = w0_func(x)
|
||
|
||
# Dirichlet BC at x=0: w(0,t) = exp(B·t)
|
||
# For short time, approximate as constant f_t ≈ 1
|
||
f_t = 1.0 # exp(B*0) = 1 for t≈0; use f_t=1 for the benchmark
|
||
|
||
t_test = 600.0 # 10 minutes
|
||
|
||
# ── Fokas method ──
|
||
print("\n--- Fokas Method ---")
|
||
for N_s in [128, 256, 512]:
|
||
t0 = time.time()
|
||
w_fokas = solve_heat_fokas(
|
||
w0, x, t_test, D,
|
||
f_t=f_t, g_L=C_robin,
|
||
bc_left='dirichlet', bc_right='robin',
|
||
N_s=N_s, contour_scale=8.0,
|
||
)
|
||
fokas_time = time.time() - t0
|
||
print(f" N_s={N_s}: {fokas_time*1000:.2f}ms")
|
||
|
||
# ── Fourier series ──
|
||
print("\n--- Fourier Series ---")
|
||
for N_terms in [100, 500, 2000]:
|
||
t0 = time.time()
|
||
w_fourier = solve_heat_fourier_series(
|
||
w0_func, L, t_test, D,
|
||
f_t=f_t, g_C=C_robin,
|
||
N_terms=N_terms, x=x,
|
||
)
|
||
fourier_time = time.time() - t0
|
||
print(f" N_terms={N_terms}: {fourier_time*1000:.2f}ms")
|
||
|
||
# ── Compare convergence ──
|
||
print("\n--- Convergence Comparison ---")
|
||
# Reference: high-resolution Fokas
|
||
w_ref = solve_heat_fokas(
|
||
w0, x, t_test, D,
|
||
f_t=f_t, g_L=C_robin,
|
||
bc_left='dirichlet', bc_right='robin',
|
||
N_s=1024, contour_scale=10.0,
|
||
)
|
||
|
||
print(f"\n {'Method':<25} {'N_pts':<8} {'RMSE vs ref':<15} {'Time (ms)':<12}")
|
||
print(f" {'-'*60}")
|
||
|
||
for N_s in [64, 128, 256, 512]:
|
||
t0 = time.time()
|
||
w_f = solve_heat_fokas(
|
||
w0, x, t_test, D,
|
||
f_t=f_t, g_L=C_robin,
|
||
bc_left='dirichlet', bc_right='robin',
|
||
N_s=N_s, contour_scale=8.0,
|
||
)
|
||
fokas_time = time.time() - t0
|
||
rmse = np.sqrt(np.mean((w_f - w_ref)**2))
|
||
print(f" {'Fokas':<25} {N_s:<8} {rmse:<15.2e} {fokas_time*1000:<12.2f}")
|
||
results.append({'method': 'fokas', 'N': N_s, 'rmse': rmse, 'time_ms': fokas_time * 1000})
|
||
|
||
for N_terms in [50, 100, 500, 1000, 2000]:
|
||
t0 = time.time()
|
||
w_f = solve_heat_fourier_series(
|
||
w0_func, L, t_test, D,
|
||
f_t=f_t, g_C=C_robin,
|
||
N_terms=N_terms, x=x,
|
||
)
|
||
fourier_time = time.time() - t0
|
||
rmse = np.sqrt(np.mean((w_f - w_ref)**2))
|
||
print(f" {'Fourier series':<25} {N_terms:<8} {rmse:<15.2e} {fourier_time*1000:<12.2f}")
|
||
results.append({'method': 'fourier', 'N': N_terms, 'rmse': rmse, 'time_ms': fourier_time * 1000})
|
||
|
||
# ── Burgers equation comparison ──
|
||
print("\n--- Burgers Equation: Hopf-Cole+Fokas vs Hopf-Cole+FFT ---")
|
||
N_burgers = 256
|
||
x_b = np.linspace(0, L, N_burgers)
|
||
u0_b = np.sin(np.pi * x_b / L) # smooth IC compatible with BCs
|
||
|
||
t0 = time.time()
|
||
u_fokas = solve_burgers_fokas(
|
||
u0_b, x_b, 100.0, nu=D,
|
||
bc_left_val=0.0, bc_right_robin_C=C_robin,
|
||
N_s=256, contour_scale=8.0,
|
||
)
|
||
fokas_burgers_time = time.time() - t0
|
||
|
||
# FFT version (periodic, for reference)
|
||
dx_b = x_b[1] - x_b[0]
|
||
t0 = time.time()
|
||
u_fft = solve_burgers_hopf_cole(u0_b, 100.0, D, dx_b)
|
||
fft_burgers_time = time.time() - t0
|
||
|
||
print(f" Hopf-Cole+Fokas: {fokas_burgers_time*1000:.2f}ms")
|
||
print(f" Hopf-Cole+FFT: {fft_burgers_time*1000:.2f}ms")
|
||
|
||
results.append({
|
||
'method': 'burgers_fokas', 'time_ms': fokas_burgers_time * 1000,
|
||
})
|
||
results.append({
|
||
'method': 'burgers_fft', 'time_ms': fft_burgers_time * 1000,
|
||
})
|
||
|
||
return results
|
||
|
||
|
||
def benchmark_all():
|
||
"""Run all benchmarks and save results."""
|
||
print("=" * 70)
|
||
print("Hopf-Cole Burgers Solver — Full Benchmark Suite")
|
||
print("Includes: FFT solver, Fokas method, Fourier series comparison")
|
||
print("=" * 70)
|
||
|
||
all_results = {}
|
||
|
||
# Part 1: FFT vs FD
|
||
print("\n\n[1/2] Hopf-Cole FFT vs Finite Difference")
|
||
fft_results = benchmark_hopf_cole()
|
||
all_results['fft_vs_fd'] = fft_results
|
||
|
||
# Part 2: Fokas vs Fourier series
|
||
print("\n\n[2/2] Fokas Method vs Fourier Series")
|
||
fokas_results = benchmark_fokas()
|
||
all_results['fokas_vs_fourier'] = fokas_results
|
||
|
||
# Save
|
||
out = {
|
||
'schema': 'hopf_cole_benchmark_v2',
|
||
'description': 'Hopf-Cole solver benchmark with Fokas unified transform method',
|
||
'reference': 'arXiv:2605.11788 (Kalimeris, Mindrinos, Paraskevopoulos 2026)',
|
||
'results': all_results,
|
||
}
|
||
out_path = Path(__file__).resolve().parent.parent.parent / 'shared-data' / 'artifacts' / 'hopf_cole_benchmark.json'
|
||
out_path.parent.mkdir(parents=True, exist_ok=True)
|
||
with open(out_path, 'w') as f:
|
||
json.dump(out, f, indent=2)
|
||
print(f"\n\nResults saved: {out_path}")
|
||
|
||
return all_results
|
||
|
||
|
||
# ── Visualization ────────────────────────────────────────────────────────────
|
||
|
||
def plot_comparison():
|
||
"""Plot Hopf-Cole vs FD solution."""
|
||
try:
|
||
import matplotlib.pyplot as plt
|
||
except ImportError:
|
||
print("matplotlib not available")
|
||
return
|
||
|
||
N = 1024
|
||
nu = 0.01
|
||
dx = 2.0 * np.pi / N
|
||
x = np.linspace(0, 2.0 * np.pi, N, endpoint=False)
|
||
u0 = np.sin(x)
|
||
|
||
fig, axes = plt.subplots(2, 2, figsize=(12, 8))
|
||
|
||
for ax, t in zip(axes.flat, [0.1, 0.3, 0.5, 1.0]):
|
||
u_hc = solve_burgers_hopf_cole(u0, t, nu, dx)
|
||
u_fd = solve_burgers_fd(u0, t, nu, dx)
|
||
|
||
ax.plot(x, u_hc, 'b-', label='Hopf-Cole (exact)', linewidth=2)
|
||
ax.plot(x, u_fd, 'r--', label='Finite Difference', linewidth=1)
|
||
ax.set_title(f't = {t}')
|
||
ax.legend()
|
||
ax.set_xlabel('x')
|
||
ax.set_ylabel('u')
|
||
|
||
plt.tight_layout()
|
||
plt.savefig('/tmp/hopf_cole_comparison.png', dpi=150)
|
||
print("Saved: /tmp/hopf_cole_comparison.png")
|
||
|
||
|
||
# ── CLI ──────────────────────────────────────────────────────────────────────
|
||
|
||
if __name__ == '__main__':
|
||
import sys
|
||
|
||
if len(sys.argv) < 2:
|
||
print("Usage: hopf_cole_burgers.py --benchmark")
|
||
print(" hopf_cole_burgers.py --benchmark-fokas")
|
||
print(" hopf_cole_burgers.py --plot")
|
||
print(" hopf_cole_burgers.py --solve <N> <nu> <t>")
|
||
sys.exit(1)
|
||
|
||
if sys.argv[1] == '--benchmark':
|
||
benchmark_all()
|
||
elif sys.argv[1] == '--benchmark-fokas':
|
||
benchmark_fokas()
|
||
elif sys.argv[1] == '--plot':
|
||
plot_comparison()
|
||
elif sys.argv[1] == '--solve':
|
||
N = int(sys.argv[2])
|
||
nu = float(sys.argv[3])
|
||
t = float(sys.argv[4])
|
||
dx = 2.0 * np.pi / N
|
||
x = np.linspace(0, 2.0 * np.pi, N, endpoint=False)
|
||
u0 = np.sin(x)
|
||
u = solve_burgers_hopf_cole(u0, t, nu, dx)
|
||
print(f"u[0:10] = {u[:10]}")
|