pt space pte solver

.gitlab/build_and_test.sh

@@ -272,6 +272,7 @@ cmake_test() {
   (
   source ${BUILD_ENV}
   export CTEST_OUTPUT_ON_FAILURE=1
+  export OMP_PROC_BIND=false
   if ${BUILD_WITH_CTEST}; then
     ctest -V -S .gitlab/build_and_test.cmake,Test,$REPORT_ERRORS
   else

CHANGELOG.md

@@ -3,6 +3,7 @@
 ## Current develop
 
 ### Added (new features/APIs/variables/...)
+- [[PR453]](https://github.com/lanl/singularity-eos/pull/453) A PT space PTE solver
 - [[PR444]](https://github.com/lanl/singularity-eos/pull/444) Add Z split modifier and electron ideal gas EOS
 
 ### Fixed (Repair bugs, etc)

doc/sphinx/src/using-closures.rst

@@ -278,6 +278,55 @@ choice of the second independent variable is discussed below and has
 implications for both the number of additional unknowns and the stability of the
 method.
 
+.. _pressure-temperature-formulation:
+The Pressure-Temperature Formulation
+`````````````````````````````````````
+
+An obvious choice is to treat the independent variables as pressure
+and temperature. Then one has only two equations and two unknowns. The
+residual contains only the volume fraction and energy summmation rules
+described above. Taylor expanding these residuals about fixed
+temeprature and pressure points leads to two residual equations of the
+form
+
+.. math::
+
+  1 - \sum_{i=0}^{N-1} f_i = (T^* - T) \sum_{i = 0}^{N-1} \left(\frac{\partial f_i}{\partial T}\right)_P + (P^* - P) \sum_{i = 0}^{N-1} \left(\frac{\partial f_i}{\partial P}\right)_T\\
+  u_{tot} - \sum_{i=0}^{N-1} u_i = (T^* - T) \sum_{i = 0}^{N-1} \left(\frac{\partial u_i}{\partial T}\right)_P + (P^* - P) \sum_{i = 0}^{N-1} \left(\frac{\partial u_i}{\partial P}\right)_T
+
+However, derivatives in the volume fraction are not easily
+accessible. To access them, we leverage the fact that
+
+.. math::
+
+  \bar{\rho}_i = \rho_i f_i,
+
+and thus
+
+.. math::
+
+  d f_i = - \frac{1}{\rho_i^2} d f_i.
+
+Thus the residual can be recast as
+
+.. math::
+
+  f_\mathrm{tot} - \sum_{i=0}^{N-1} = (T^* - T) \sum_{i = 0}^{N-1} \frac{\bar{\rho}_i}{\rho_i^2} \left(\frac{\partial \rho_i}{\partial T}\right)_P + (P^* - P) \sum_{i = 0}^{N-1} \frac{\bar{\rho}_i}{\rho_i^2} \left(\frac{\partial \rho_i}{\partial P}\right)_T\\
+  u_\mathrm{tot} - u_i = (T^* - T) \sum_{i = 0}^{N-1} \left(\frac{\partial u_i}{\partial T}\right)_P + (P^* - P) \sum_{i = 0}^{N-1} \left(\frac{\partial u_i}{\partial P}\right)_T
+
+where :math:`\rho_{\mathrm{tot}}` is the sum of densities over all
+materials. These residual equations can then be cast as a matrix
+equation to solve for pressure and temperature.
+
+The primary advantage of the pressure-temperature space solver is that
+it has only two independent variables and two unknowns, meaning the
+cost scales only linearly with the number of materials, not
+quadratically (or worse). The primary disadvantage, is that most
+equations of state are not formulated in terms of pressure and
+temperature, meaning additional inversions are required.
+
+In the code, this method is referred to as ``PTESolverPT``.
+
 .. _density-energy-formalism:
 The Density-Energy Formulation
 ''''''''''''''''''''''''''''''