Faster CircularResistiveWall

Update CircularResistiveWall class in resistive_wall.py to replace numerical integrations with an analytical forms for improved performance.

• Numerical integrations are removed.
• LongWakeExact and TransWakeExact are used even in the case where time > 20*t0.
• "atol" for LongWakeExact is removed.

I believe "atol" is not needed when considering the fundamental theorem of beam loading. This is because Wl(0) = Wl(0+)/2 is comes from the equation Wl(0) = [Wl(0-)*l/2 + Wl(0+)*l/2] / l, which represents the average Wl value of a very narrow bin (as the length of bin l -> 0) at the origin. Defining "atol" for the fundamental theorem of beam loading introduces the possibility that more than one point could be divided by 2, which I believe is mathematically and physically unacceptable. Thus, it is correct to consider the fundamental theorem of beam loading only at the origin. PyHEADTAIL also considers the fundamental theorem of beam loading only at the origin. Please see def function_longitudinal at the link (line 341).

mbtrack2/impedance/resistive_wall.py

@@ -5,7 +5,7 @@ Define resistive wall elements based on the WakeField class.
 
 import numpy as np
 from scipy.constants import c, epsilon_0, mu_0
-from scipy.special import wofz
+from scipy.special import wofz as _scipy_wofz
 
 from mbtrack2.impedance.wakefield import Impedance, WakeField, WakeFunction
 
@@ -238,25 +238,39 @@ class CircularResistiveWall(WakeField):
             idx2 = np.logical_not(idx1)
             wt[idx2] = self.__TransWakeApprox(time[idx2])
         return wt
+    
+    def __wofz(self, z):
+        """
+        Compute the Faddeeva function w(z) = exp(-z**2) * erfc(-i*z).
+
+        Returns
+        -------
+        tuple
+            Real and imaginary parts of the Faddeeva function.
+        """
+        res = _scipy_wofz(z)
+        return res.real, res.imag
 
     def __LongWakeExact(self, t, factor):
+        w1re, _ = self.__wofz( 1j * np.sqrt(2 * t / self.t0) )
+        w2re, _ = self.__wofz( np.exp(1j * np.pi / 6) *
+                                np.sqrt(2 * t / self.t0) )
+
         wl = factor * ( 4 * np.exp(-1 * t / self.t0) *
              np.cos(np.sqrt(3) * t / self.t0)
-             + np.real( wofz( 1j *
-                        np.sqrt(2 * t / self.t0) ) )
-             - 2 * np.real( wofz( np.exp(1j * np.pi / 6) *
-                            np.sqrt(2 * t / self.t0) ) ) )
+             + w1re - 2 * w2re )
         return wl
 
     def __TransWakeExact(self, t, factor):
+        w1re, _ = self.__wofz( 1j * np.sqrt(2 * t / self.t0) )
+        w2re, w2im = self.__wofz( np.exp(1j * np.pi / 6) *
+                                    np.sqrt(2 * t / self.t0) )
+
         wt = factor * ( 2 * np.exp(-1 * t / self.t0) *
              ( np.sqrt(3) * np.sin(np.sqrt(3) * t / self.t0)
-               - np.cos(np.sqrt(3) * t / self.t0) )
-             + np.real( wofz( 1j *
-                        np.sqrt(2 * t / self.t0) ) )
-             + 2 * np.real( np.exp(-1j * np.pi / 3) *
-                            wofz( np.exp(1j * np.pi / 6) *
-                            np.sqrt(2 * t / self.t0) ) ) )
+             - np.cos(np.sqrt(3) * t / self.t0) )
+             + w1re + 2 * ( np.cos(-np.pi/3) * w2re
+             - np.sin(-np.pi/3) * w2im ) )
         return wt
 
     def __LongWakeApprox(self, t):