Correct external process algebraic rows jacobian

[K20shores]

Closes #1094

Before this PR, external model processes added their jacobian terms (SubtractJacobianTerms) to the rows of algebraic variables, and the constraints (SubtractConstraintJacobian) then added their terms on top. However, the row of an algebraic variable in the jacobian should hold only the constraint's Jacobian terms. After this PR, micm still calls SubtractJacobianTerms for all of the rows that a process affects, but it removes the entries in algebraic rows before the constraints add their terms.

In math:

Split the state into two sets of indices, $D$ is the set of differential variables, and $A$ is the set of algebraic variables. Below, $M$ is the mass matrix (defined in solver builder in micm), $y$ is the state variables (concentrations, number concentration, radius, charge, etc), $y'$ is the time derivative of each variable. Without constraints $M=I$, $I$ being the identity matrix, and then $y'$ is just the forcing.

$$F(y)=My', \quad M = \mathrm{diag}(m), \quad m_i = 1 \;\; \forall\, i \in D, \quad m_i = 0 \;\; \forall\, i \in A,$$

We can describe each process that can affect the jacobian or forcing function as one of three things

• $r(y)$: the built-in reaction rates
• $p(y)$: external model process rates
• $g(y)$: the constraint residuals. There is one residual $\forall i \in A$

The correct right-hand sides of the forcing uses the rate only in differential rows and the constraints only in algebraic rows

$$\begin{aligned} F_i(y) &= r_i(y) + p_i(y) \quad &\forall i\in D \\\ F_i(y) &= g_i(y) \quad &\forall i\in A \end{aligned}$$

and then the correct jacobian similarly uses only the constraints in algebraic rows

$$\begin{aligned} J_i(y) &= \partial r_i / \partial y + \partial p_i / \partial y \quad &\forall i\in D \\\ J_i(y) &= \partial g_i / \partial y \quad &\forall i\in A \end{aligned}$$

Before this PR, we were incorrectly computing the jacobian on algebraic rows as

$$J_i(y) = \partial g_i / \partial y + \partial p_i / \partial y \quad \forall i\in A$$

After this PR, we have the corrected jacobian

$$\begin{aligned} J_i(y) &= \partial r_i / \partial y + \partial p_i / \partial y \quad &\forall i\in D \\\ J_i(y) &= \partial g_i / \partial y \quad &\forall i\in A \end{aligned}$$

[codecov-commenter]

Codecov Report

❌ Patch coverage is 97.05882% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 96.14%. Comparing base (f2b7b0a) to head (adcaaaa).

Files with missing lines	Patch %	Lines
include/micm/solver/external_model_dispatcher.hpp	96.15%	1 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #1095   +/-   ##
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  Coverage   96.13%   96.14%           
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  Files          58       58           
  Lines        4791     4825   +34     
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+ Hits         4606     4639   +33     
- Misses        185      186    +1     

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[dwfncar]

All OK