Applications - Flow over wings - planes - birds - insects - 2D viscous flows - Fluid mixing - How jellyfish eat - pollutants in ocean - Planetary transport - Newton's laws for each celestial body - Inverted pendulum - Rockets - Disease modeling Approaches: - write equations of motion as [[Partial Differential Equations (PDEs)|PDEs]] - create computational domain of cells, discretize PDEs into [[Ordinary Differential Equations (ODEs)|ODEs]] ## Example Seattle weather Three types of weather: 1. Rainy 2. Cloudy 3. Nice If rainy or cloudy, 50% it'll be the same conditions tomorrow, 25% nice, 25% the other If nice, 50% it will be rainy or cloudy tomorrow Represent as vectors Today: $ \mathbf{x}_{today}=\begin{bmatrix}p_{rainy}\\p_{nice}\\p_{cloudy}\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix} $ Tomorrow: $ \mathbf{x}_{tomorrow}=\mathbf{A}\mathbf{x}_{atoday} $ Create model: $ \mathbf{A}=\begin{bmatrix}p_{rainy,\,rainy}&p_{nice,\,rainy}&p_{cloudy,\,rainy}\\p_{rainy,\,nice}&p_{nice,\,nice}&p_{cloudy,\,nice}\\p_{rainy,\, cloudy}&p_{nice,\,cloudy}&p_{cloudy,\,cloudy}\end{bmatrix}=\begin{bmatrix}0.5&0.5&0.25\\0.25&0&0.25\\0.25&0.5&0.5\end{bmatrix} $ $ \mathbf{x}_{tomorrow}=\begin{bmatrix}0.5\\0.25\\0.25\end{bmatrix} $