Mathematical formulation of the reaction-transport problem

The PALEO models define various special cases of a general DAE problem (these can be combined, providing the number of implicit state variables $S_{impl}$ is equal to the number of algebraic constraints $G$ plus the number of total variables $U$):

Explicit ODE

The time derivative of explicit state variables $S_{explicit}$ (a subset of all state variables $S_{all}$) are an explicit function of time $t$:

\[\frac{dS_{explicit}}{dt} = F(S_{all}, t)\]

where explicit state variables $S_{explicit}$ are identified by PALEO attribute :vfunction = PALEOboxes.VF_StateExplicit and paired time derivatives $F$ by :vfunction = PALEOboxes.VF_Deriv along with the naming convention <statevarname>, <statevarname>_sms.

Algebraic constraints

State variables $S_{impl}$ (a subset of all state variables $S_{all}$) are defined by algebraic constraints $G$:

\[0 = G(S_{all}, t)\]

where implicit state variables $S_{impl}$ are identified by PALEO attribute :vfunction = PALEOboxes.VF_State and algebraic constaints $G$ by :vfunction = PALEOboxes.VF_Constraint (these are not paired).

ODE with variable substitution

State variables $S_{impl}$ (a subset of all state variables $S_{all}$) are defined the time evolution of total variables $U(S_{all})$ (this case is common in biogeochemistry where the total variables $U$ represent conserved chemical elements, and the state variables eg primary species):

\[\frac{dU(S_{all})}{dt} = F(U(S_{all}), t)\]

where total variables $U$ are identified by PALEO attribute :vfunction = PALEOboxes.VF_Total and paired time derivatives $F$ by :vfunction = PALEOboxes.VF_Deriv along with the naming convention <totalvarname>, <totalvarname>_sms, and implicit state variables $S_{impl}$ are identified by PALEO attribute :vfunction = PALEOboxes.VF_StateTotal.