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

@@ -45,7 +45,8 @@ class CircularResistiveWall(WakeField):
     Resistive wall WakeField element for a circular beam pipe.
     
     Impedance from approximated formulas from Eq. (2.77) of Chao book [1].
-    Wake function formulas from [2, 3].
+    Wake function formulas from [2, 3] and the fundamental theorem of
+    beam loading from [4].
         
     Parameters
     ----------
@@ -67,7 +68,7 @@ class CircularResistiveWall(WakeField):
     ----------
     [1] : Chao, A. W. (1993). Physics of collective beam instabilities in high 
     energy accelerators. Wiley.
-    [2] : Ivanyan, Mikayel I. and Tsakanov, Vasili M. "Analytical treatment of
+    [2] : Ivanyan, Mikayel I., and Vasili M. Tsakanov. "Analytical treatment of
     resistive wake potentials in round pipes." Nuclear Instruments and Methods
     in Physics Research Section A: Accelerators, Spectrometers, Detectors and
     Associated Equipment 522, no. 3 (2004): 223-229.
@@ -75,6 +76,9 @@ class CircularResistiveWall(WakeField):
     interbunch collective beam motions in storage rings." Nuclear Instruments 
     and Methods in Physics Research Section A: Accelerators, Spectrometers, 
     Detectors and Associated Equipment 806 (2016): 221-230.
+    [4] : Zotter, Bruno W., and Semyon A. Kheifets (1998). Impedances and wakes
+    in high-energy particle accelerators. World Scientific.
+
 
     """
 
@@ -129,9 +133,9 @@ class CircularResistiveWall(WakeField):
         The approxmiated expression is valid if the time is large compared
         to the characteristic time t0.
         
-        Eq. (11) in [1] is completely equivalent to Eq. (22) in [2].
+        Eq. (11) in [1] is completely identical to Eq. (22) in [2].
         
-        The fundamental theorem  of beam loading is applied for the exact
+        The fundamental theorem of beam loading [3] is applied for the exact
         expression of the longitudinal wake function: Wl(0) = Wl(0+)/2.
 
         Parameters
@@ -148,7 +152,7 @@ class CircularResistiveWall(WakeField):
             
         References
         ----------
-        [1] : Ivanyan, Mikayel I. and Tsakanov, Vasili M. "Analytical treatment of
+        [1] : Ivanyan, Mikayel I., and Vasili M. Tsakanov. "Analytical treatment of
         resistive wake potentials in round pipes." Nuclear Instruments and Methods
         in Physics Research Section A: Accelerators, Spectrometers, Detectors and
         Associated Equipment 522, no. 3 (2004): 223-229.
@@ -156,6 +160,8 @@ class CircularResistiveWall(WakeField):
         interbunch collective beam motions in storage rings." Nuclear Instruments 
         and Methods in Physics Research Section A: Accelerators, Spectrometers, 
         Detectors and Associated Equipment 806 (2016): 221-230.
+        [3] : Zotter, Bruno W., and Semyon A. Kheifets (1998). Impedances and wakes
+        in high-energy particle accelerators. World Scientific.
         """
         wl = np.zeros_like(time)
         idx1 = time < 0
@@ -179,7 +185,7 @@ class CircularResistiveWall(WakeField):
         The approxmiated expression is valid if the time is large compared
         to the characteristic time t0.
         
-        Eq. (11) in [1] is completely equivalent to Eq. (25) in [2].
+        Eq. (11) in [1] is completely identical to Eq. (25) in [2].
         Corrected the typos in the last two terms in exact expression
         (Eq. (11), of [1]).
 
@@ -193,11 +199,11 @@ class CircularResistiveWall(WakeField):
         Returns
         -------
         wt : array of float
-            Transverse wake function in [V/C].
+            Transverse wake function in [V/C/m].
             
         References
         ----------
-        [1] : Ivanyan, Mikayel I. and Tsakanov, Vasili M. "Analytical treatment of
+        [1] : Ivanyan, Mikayel I., and Vasili M. Tsakanov. "Analytical treatment of
         resistive wake potentials in round pipes." Nuclear Instruments and Methods
         in Physics Research Section A: Accelerators, Spectrometers, Detectors and
         Associated Equipment 522, no. 3 (2004): 223-229.