Update the e-Ar(1,r) collision integral fits (#298)

The original fits (in functions collision::argon::eAr11,
collision::argon::eAr12 etc) cause problems in the third order
electron thermal conductivity calculation.  Specifically, although the
fits aren't that bad, the resulting approximation of the quantity

L11 * L22 - L12 * L12,

(where L here corresponds the \Lambda in the usual notation) behaves
very badly (e.g., it goes negative near 8000K).  This bad behavior can
cause the electron thermal conductivity to also behave badly.

This commit updates the fits to eliminate this problem.

Due to these changes, a number of regression tests fail at this
commit.  These will be fixed in subsequent commits.

src/collision_integrals.cpp

@@ -141,59 +141,63 @@ MFEM_HOST_DEVICE double ArAr1P11(const double &T) {
   return 4.574321e-18 * pow(T, -0.1805);
 }
 
+// General form of the fit used for the e-Ar collision integrals
+MFEM_HOST_DEVICE double logT_fit(const double logT, const double coeff[9]) {
+  // Evaluate powers of log(T) from -1 through 7
+  double logT_power[9];
+  logT_power[0] = 1. / logT;
+  logT_power[1] = 1.;
+  for (int k = 0; k < 7; k++) {
+    logT_power[k + 2] = logT_power[k + 1] * logT;
+  }
+
+  // fit = sum_k c_k (log(T)^k) for k=-1 to 7
+  double fit = 0.0;
+  for (int k = 0; k < 9; k++) {
+    fit += coeff[k] * logT_power[k];
+  }
+
+  return fit;
+}
+
 /*
   e-Ar (l,r) are fitted over numerical quadrature of definitions.
   Q_{e,Ar}^(1), elastic momentum transfer cross section, is determined by a 7-parameter shifted MERT model,
   fitted over BSR LXCat dataset.
 */
 MFEM_HOST_DEVICE double eAr11(const double &T) {
   const double logT = log(T);
-  if (T < 1.2e4) {
-    return 5.8664e-22 * logT * logT * logT - 6.3417e-21 * logT * logT + 3.2083e-21 * logT + 9.0686e-20;
-  } else {
-    double tmp = logT - 10.9082;
-    return 12.1818e-20 * exp(-0.4186 * tmp * tmp) + 8.6949e-22;
-  }
+  const double coeff[9] = {6.36254140e-18, 1.84835040e-18,  -5.87727093e-18, 3.20023027e-18, -8.50509054e-19,
+                           1.28163820e-19, -1.11712910e-20, 5.25649382e-22,  -1.03296658e-23};
+  return logT_fit(logT, coeff);
 }
 
 MFEM_HOST_DEVICE double eAr12(const double &T) {
   const double logT = log(T);
-  if (T < 1.0e4) {
-    return 5.0435e-22 * logT * logT * logT - 4.0041e-21 * logT * logT - 1.3234e-20 * logT + 1.1966e-19;
-  } else {
-    double tmp = logT - 10.5348;
-    return 12.5836e-20 * exp(-0.5116 * tmp * tmp) + 8.7254e-22;
-  }
+  const double coeff[9] = {1.91338172e-17, 5.45418129e-18,  -1.78361685e-17, 9.75657946e-18, -2.61115722e-18,
+                           3.98310268e-19, -3.53503678e-20, 1.70375066e-21,  -3.45211955e-23};
+  return logT_fit(logT, coeff);
 }
 
 MFEM_HOST_DEVICE double eAr13(const double &T) {
   const double logT = log(T);
-  if (T < 8.2e3) {
-    return 4.3150e-22 * logT * logT * logT - 2.1312e-21 * logT * logT - 2.5311e-20 * logT + 1.3866e-19;
-  } else {
-    double tmp = logT - 10.2802;
-    return 12.9711e-20 * exp(-0.5725 * tmp * tmp) + 1.6371e-21;
-  }
+  const double coeff[9] = {3.04685398e-17, 8.39750994e-18,  -2.88132528e-17, 1.60147037e-17, -4.34837891e-18,
+                           6.73136845e-19, -6.06704580e-20, 2.97216168e-21,  -6.12760944e-23};
+  return logT_fit(logT, coeff);
 }
 
 MFEM_HOST_DEVICE double eAr14(const double &T) {
   const double logT = log(T);
-  if (T < 7.1e3) {
-    return 3.9545e-22 * logT * logT * logT - 1.1198e-21 * logT * logT - 3.1302e-20 * logT + 1.4507e-19;
-  } else {
-    double tmp = logT - 10.0853;
-    return 13.2903e-20 * exp(-0.6150 * tmp * tmp) + 1.7854e-21;
-  }
+  const double coeff[9] = {3.90777949e-17, 1.04696956e-17,  -3.73774204e-17, 2.10610498e-17, -5.79029566e-18,
+                           9.07573157e-19, -8.28466766e-20, 4.11188110e-21,  -8.59225098e-23};
+  return logT_fit(logT, coeff);
 }
 
 MFEM_HOST_DEVICE double eAr15(const double &T) {
   const double logT = log(T);
-  if (T < 6.0e3) {
-    return 2.8521e-22 * logT * logT * logT + 9.9567e-22 * logT * logT - 4.2614e-20 * logT + 1.6026e-19;
-  } else {
-    double tmp = logT - 9.9275;
-    return 13.4901e-20 * exp(-0.6295 * tmp * tmp) + 4.6041e-22;
-  }
+  const double coeff[9] = {4.41333290e-17, 1.15696010e-17,  -4.25651305e-17, 2.42442440e-17, -6.73359258e-18,
+                           1.06641697e-18, -9.83933863e-20, 4.93775812e-21,  -1.04362372e-22};
+  return logT_fit(logT, coeff);
 }
 
 }  // namespace argon