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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.