16.2 Example: chemical kinetics

As an example of a system of DAEs, consider following chemical kinetics problem(Robertson 1966). The nondimensionalized DAE consists of two differential equations and one algebraic constraint. The differential equations describe the reactions from reactants \(y_1\) and \(y_2\) to the product \(y_3\), and the algebraic equation describes the mass conservation. (Serban and Hindmarsh 2021).

\[ \frac{dy_1}{dt} + \alpha y_1 - \beta y_2 y_3 = 0 \frac{dy_2}{dt} - \alpha y_1 + \beta y_2 y_3 + \gamma y_2^2 = 0 y_1 + y_2 + y_3 - 1.0 = 0 \]

The state equations implicitly defines the state \((y_1(t), y_2(t), y_3(t))\) at future times as a function of an initial state and the system parameters, in this example the reaction rate coefficients \((\alpha, \beta, \gamma)\).

Unlike solving ODEs, solving DAEs requires a consistent initial condition. That is, one must specify both \(y(t_0)\) and \(y'(t_0)\) so that residual function becomes zero at initial time \(t_0\) \[ r(y'(t_0), y(t_0), t_0) = 0 \]

References

Robertson, H. H. 1966. “The Solution of a Set of Reaction Rate Equations.” In Numerical Analysis, an Introduction, 178–82. Lodon; New York: Academic Press.
Serban, Radu, and Alan C. Hindmarsh. 2021. “Example Programs for IDAS.” LLNL-TR-437091. Lawrence Livermore National Laboratory.