Add clusters/solver.py to solve the Fokker-Planck equation

Adopt the finite difference scheme to solve the Fokker-Planck equation,
following Park & Petrosian (1996, ApJS, 103, 255).

1 files changed, 276 insertions, 0 deletions

diff --git a/fg21sim/extragalactic/clusters/solver.py b/fg21sim/extragalactic/clusters/solver.py
new file mode 100644
index 0000000..5078c5e
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+++ b/fg21sim/extragalactic/clusters/solver.py
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+# Copyright (c) 2017 Weitian LI <liweitianux@live.com>
+# MIT license
+
+"""
+Solve the Fokker-Planck equation to derive the time evolution
+of the electron spectrum (or number density distribution).
+"""
+
+import logging
+
+import numpy as np
+
+
+logger = logging.getLogger(__name__)
+
+
+def TDMAsolver(a, b, c, d):
+    """
+    Tri-diagonal matrix algorithm (a.k.a Thomas algorithm) solver,
+    which is much faster than the generic Gaussian elimination algorithm.
+
+    a[i]*x[i-1] + b[i]*x[i] + c[i]*x[i+1] = d[i],
+    where: a[1] = c[n] = 0
+
+    References
+    ----------
+    [1] http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
+
+    Credit
+    ------
+    [1] https://gist.github.com/cbellei/8ab3ab8551b8dfc8b081c518ccd9ada9
+    """
+    # Number of equations
+    nf = len(d)
+    # Copy the input arrays
+    ac, bc, cc, dc = map(np.array, (a, b, c, d))
+    for it in range(1, nf):
+        mc = ac[it-1] / bc[it-1]
+        bc[it] -= mc*cc[it-1]
+        dc[it] -= mc*dc[it-1]
+
+    xc = bc
+    xc[-1] = dc[-1] / bc[-1]
+
+    for il in range(nf-2, -1, -1):
+        xc[il] = (dc[il] - cc[il]*xc[il+1]) / bc[il]
+
+    return xc
+
+
+class FokkerPlanckSolver:
+    """
+    Solve the Fokker-Planck equation.
+
+    ∂u(x,t)   ∂  /                ∂u(x) \
+    ------- = -- | B(x)u(x) + C(x)----- | + Q(x,t)
+       ∂t     ∂x \                  ∂x  /
+
+    u(x,t) : distribution/spectrum w.r.t. x at different times
+    B(x,t) : advection coefficient
+    C(x,t) : diffusion coefficient (>=0)
+    Q(x,t) : injection coefficient (>=0)
+
+    References
+    ----------
+    [1] Park & Petrosian 1996, ApJS, 103, 255
+        http://adsabs.harvard.edu/abs/1996ApJS..103..255P
+    [2] Donnert & Brunetti 2014, MNRAS, 443, 3564
+        http://adsabs.harvard.edu/abs/2014MNRAS.443.3564D
+    """
+
+    def __init__(self, xmin, xmax, grid_num, buffer_np, tstep,
+                 f_advection, f_diffusion, f_injection):
+        self.xmin = xmin
+        self.xmax = xmax
+        # Number of points on the logarithmic grid
+        self.grid_num = grid_num
+        # Number of grid points for the buffer region near the lower boundary
+        self.buffer_np = buffer_np
+        # Time step
+        self.tstep = tstep
+        # Function f(x,t) to calculate the advection coefficient B(x,t)
+        self.f_advection = f_advection
+        # Function f(x,t) to calculate the diffusion coefficient C(x,t)
+        self.f_diffusion = f_diffusion
+        # Function f(x,t) to calculate the injection coefficient Q(x,t)
+        self.f_injection = f_injection
+
+    @property
+    def x(self):
+        """
+        X values of the adopted logarithmic grid.
+        """
+        grid = np.logspace(np.log10(self.xmin), np.log10(self.xmax),
+                           num=self.grid_num)
+        return grid
+
+    @property
+    def dx(self):
+        """
+        Values of delta X on the grid.
+
+        dx[i] = (x[i+1] - x[i-1]) / 2
+        Thus the first and last element is NaN.
+
+        References: Ref.[1],Eq.(8)
+        """
+        x = self.x
+        dx_ = (x[2:] - x[:-2]) / 2
+        grid = np.concatenate([[np.nan], dx_, [np.nan]])
+        return grid
+
+    @property
+    def dx_phalf(self):
+        """
+        Values of dx[i+1/2] on the grid.
+
+        dx[i+1/2] = x[i+1] - x[i]
+        Thus the last element is NaN.
+
+        References: Ref.[1],Eq.(8)
+        """
+        x = self.x
+        dx_ = x[1:] - x[:-1]
+        grid = np.concatenate([dx_, [np.nan]])
+        return grid
+
+    @property
+    def dx_mhalf(self):
+        """
+        Values of dx[i-1/2] on the grid.
+
+        dx[i-1/2] = x[i] - x[i-1]
+        Thus the first element is NaN.
+        """
+        x = self.x
+        dx_ = x[1:] - x[:-1]
+        grid = np.concatenate([[np.nan], dx_])
+        return grid
+
+    @staticmethod
+    def X_phalf(X):
+        """
+        Calculate the values at midpoints (+1/2) for the given quantity.
+
+        X[i+1/2] = (X[i] + X[i+1]) / 2
+        Thus the last element is NaN.
+
+        References: Ref.[1],Eq.(10)
+        """
+        Xmid = (X[1:] + X[:-1]) / 2
+        return np.concatenate([Xmid, [np.nan]])
+
+    @staticmethod
+    def X_mhalf(X):
+        """
+        Calculate the values at midpoints (-1/2) for the given quantity.
+
+        X[i-1/2] = (X[i-1] + X[i]) / 2
+        Thus the first element is NaN.
+        """
+        Xmid = (X[1:] + X[:-1]) / 2
+        return np.concatenate([[np.nan], Xmid])
+
+    @staticmethod
+    def W(w):
+        # References: Ref.[1],Eqs.(27,35)
+        w = np.abs(w)
+        W = np.zeros(w.shape) * np.nan
+        mask = (w < 0.1)
+        W[mask] = 1.0 / (1 + w[mask]**2/24 + w[mask]**4/1920)
+        W[~mask] = (w[~mask] * np.exp(-w[~mask]/2) /
+                    (1 - np.exp(-w[~mask])))
+        return W
+
+    def tridiagonal_coefs(self, tc, uc):
+        """
+        Calculate the coefficients for the tridiagonal system of linear
+        equations corresponding to the original Fokker-Planck equation.
+
+        -a[i]*u[i-1] + b[i]*u[i] - c[i]*u[i+1] = r[i],
+        where: a[0] = c[N-1] = 0
+
+        NOTE
+        ----
+        When i=0 or i=N-1, b[i] is invalid due to X[-1/2] or X[N-1/2] are
+        invalid. Therefore, b[0] and b[N-1] should be alternatively
+        calculated with (e.g., no-flux) boundary condition considered.
+
+        References: Ref.[1],Eqs.(16,18,34)
+        """
+        x = self.x
+        dx = self.dx
+        dx_phalf = self.dx_phalf
+        dx_mhalf = self.dx_mhalf
+        dt = self.tstep
+        B = np.array([self.f_advection(x_, tc) for x_ in x])
+        C = np.array([self.f_diffusion(x_, tc) for x_ in x])
+        Q = np.array([self.f_injection(x_, tc) for x_ in x])
+        #
+        B_phalf = self.X_phalf(B)
+        B_mhalf = self.X_mhalf(B)
+        C_phalf = self.X_phalf(C)
+        C_mhalf = self.X_mhalf(C)
+        w_phalf = dx_phalf * B_phalf / C_phalf
+        w_mhalf = dx_mhalf * B_mhalf / C_mhalf
+        W_phalf = self.W(w_phalf)
+        W_mhalf = self.W(w_mhalf)
+        Wplus_phalf = W_phalf * np.exp(w_phalf/2)
+        Wplus_mhalf = W_mhalf * np.exp(w_mhalf/2)
+        Wminus_phalf = W_phalf * np.exp(-w_phalf/2)
+        Wminus_mhalf = W_mhalf * np.exp(-w_mhalf/2)
+        #
+        a = (dt/dx) * (C_mhalf/dx_mhalf) * Wminus_mhalf
+        a[0] = 0.0  # Fix a[0] which is NaN
+        c = (dt/dx) * (C_phalf/dx_phalf) * Wplus_phalf
+        c[-1] = 0.0  # Fix c[-1] which is NaN
+        b = 1 + (dt/dx) * ((C_mhalf/dx_mhalf) * Wplus_mhalf +
+                           (C_phalf/dx_phalf) * Wminus_phalf)
+        # Calculate b[0] & b[-1], considering the no-flux boundary condition
+        b[0] = 1 + (dt/dx[0]) * (C_phalf[0]/dx_phalf[0])*Wminus_phalf[0]
+        b[-1] = 1 + (dt/dx[-1]) * (C_mhalf[-1]/dx_mhalf[-1])*Wplus_mhalf[-1]
+        r = dt * Q + uc
+        return (a, b, c, r)
+
+    def fix_boundary(self, uc):
+        """
+        Truncate the lower end (i.e., near the lower boundary) of the
+        distribution/spectrum and then extrapolate as a power law, in order
+        to avoid the unphysical pile-up of electrons at the lower regime.
+
+        References: Ref.[2],Sec.(3.3)
+        """
+        uc = np.array(uc)
+        x = self.x
+        # Calculate the power-law index
+        xa = x[self.buffer_np]
+        xb = x[self.buffer_np+1]
+        ya = uc[self.buffer_np]
+        yb = uc[self.buffer_np+1]
+        s = np.log(yb/ya) / np.log(xb/xa)
+        uc[:self.buffer_np] = ya * (x[:self.buffer_np] / xa) ** s
+        return uc
+
+    def solve_step(self, tc, uc):
+        """
+        Solve the Fokker-Planck equation by a single step.
+        """
+        a, b, c, r = self.tridiagonal_coefs(tc, uc)
+        TDM_a = -a[1:]  # Also drop the first element
+        TDM_b = b
+        TDM_c = -c[:-1]  # Also drop the last element
+        TDM_rhs = r
+        t2 = tc + self.tstep
+        u2 = TDMAsolver(TDM_a, TDM_b, TDM_c, TDM_rhs)
+        u2 = self.fix_boundary(u2)
+        # Clear negative number densities
+        # u2[u2 < 0] = 0
+        return (t2, u2)
+
+    def solve(self, u0, tstart, tstop):
+        """
+        Solve the Fokker-Planck equation from ``tstart`` to ``tstop``,
+        with initial spectrum/distribution ``u0``.
+        """
+        uc = u0
+        tc = tstart
+        logger.info("Solving Fokker-Planck equation: " +
+                    "time: %.3f - %.3f" % (tstart, tstop))
+        nstep = (tstop - tc) / self.tstep
+        i = 0
+        while tc < tstop:
+            i += 1
+            logger.info("[%d/%d] t=%.3f ..." % (i, nstep, tc))
+            tc, uc = self.solve_step(tc, uc)
+        return uc