10.3 Differential-Algebraic equation (DAE) solver
Stan provides two higher order functions for solving initial value problems specified as Differential-Algebraic Equations (DAEs) with index-1 (Serban et al. 2021).
Solving an initial value DAE means given a set of residual functions \(r(y'(t, \theta), y(t, \theta), t)\) and initial conditions \((y(t_0, \theta), y'(t_0, \theta))\), solving for \(y\) at a sequence of times \(t_0 < t_1 \leq t_2, \cdots \leq t_n\). The residual function \(r(y', y, t, \theta)\) will be defined as a function with a certain signature and provided along with the initial conditions and output times to one of the DAE solver functions.
Similar to ODE solvers, the DAE solver function takes extra arguments
that are passed along unmodified to the user-supplied system function.
Because there can be any number of these arguments and they can be of different types,
they are denoted below as ..., and the types of these arguments,
also represented by ... in the DAE solver call, must match the types of the arguments represented by
... in the user-supplied system function.
10.3.1 The DAE solver
array[] vector dae(function residual, vector initial_state, vector initial_state_derivative, real initial_time, array[] real times, ...)
Solves the DAE system using the backward differentiation formula (BDF)
method (Serban et al. 2021).
Available since 2.29
array[] vector dae_tol(function residual, vector initial_state, vector initial_state_derivative, real initial_time, array[] real times, data real rel_tol, data real abs_tol, int max_num_steps, ...)
Solves the DAE system for the times provided using the backward differentiation formula (BDF) method with additional control
parameters for the solver.
Available since 2.29
10.3.2 DAE system function
The first argument to the DAE solver is the DAE residual
function. The DAE residual function must have a vector return type, and the
first three arguments must be a real, vector, and vector, in that order. These three
arguments are followed by the variadic arguments that are passed through from
the DAE solver function call:
vector residual(real time, vector state, vector state_derivative, ...)The DAE residual function should return the residuals at the time and state provided. The length of the returned vector must match the length of the state input into the function.
The arguments to this function are:
time, the time to evaluate the DAE systemstate, the state of the DAE system at the time specifiedstate_derivative, the time derivatives of the state of the DAE system at the time specified..., sequence of arguments passed unmodified from the DAE solve function call. The types here must match the types in the...arguments of the DAE solve function call.
10.3.3 Arguments to the DAE solver
The arguments to the DAE solver are
residual: DAE residual function,initial_state: initial state, typevector,initial_state_derivative: time derivative of the initial state, typevector,initial_time: initial time, typereal,times: solution times, typearray[] real,...: sequence of arguments that will be passed through unmodified to the DAE residual function. The types here must match the types in the...arguments of the DAE residual function.
For dae_tol, the following three
parameters must be provided after times and before the ... arguments:
datarel_tol: relative tolerance for the DAE solver, typereal, data only,dataabs_tol: absolute tolerance for the DAE solver, typereal, data only, andmax_num_steps: maximum number of steps to take between output times in the DAE solver, typeint, data only.
Because the tolerances are data arguments, they must be supplied as
primitive numerics or defined in either the data
or transformed data blocks. They cannot be parameters, transformed parameters
or functions of parameters or transformed parameters.
10.3.3.1 Consistency of the initial conditions
The user is responsible to ensure
the residual function becomes zero at the initial time, t0, when the
arguments initial_state and initial_state_derivative are
introduced as state and state_derivative, respectively.
10.3.3.2 Return values
The return value for the DAE solvers is an array of vectors (type array[] vector),
one vector representing the state of the system at every time specified in
the times argument.
10.3.3.3 Array and vector sizes
The sizes must match, and in particular, the following groups are of the same size:
state variables and state derivatives passed into the residual function, the residual returned by the residual function, initial state and initial state derivatives passed into the solver, and length of each vector in the output,
number of solution times and number of vectors in the output.