16 Differential-Algebraic Equations
Stan support solving systems of differential-algebraic equations (DAEs) of index 1 (Serban et al. 2021). The solver adaptively refines the solutions in order to satisfy given tolerances.
One can think a differential-algebraic system of equations as ODEs with additional algebraic constraits applied to some of the variables. In such a system, the variable derivatives may not be expressed explicitly with a right-hand-side as in ODEs, but implicitly constrained.
Similar to ODE solvers, the DAE solvers must not only provide the solution to the DAE itself, but also the gradient of the DAE solution with respect to parameters (the sensitivities). Stan’s DAE solver uses the forward sensitivity calculation to expand the base DAE system with additional DAE equations for the gradients of the solution. For each parameter, an additional full set of \(N\) sensitivity states are added meaning that the full DAE solved has \[N \, + N \cdot M\] states.
Two interfaces are provided for the forward sensitivity solver: one with default tolerances and default max number of steps, and one that allows these controls to be modified. Choosing tolerances is important for making any of the solvers work well – the defaults will not work everywhere. The tolerances should be chosen primarily with consideration to the scales of the solutions, the accuracy needed for the solutions, and how the solutions are used in the model. The same principles in the control parameters section apply here.
Internally Stan’s DAE solver uses a variable-step, variable-order, backward-differentiation formula implementation (Cohen and Hindmarsh 1996; Serban and Hindmarsh 2005).