Semi-Implicit Method for Pressure-Linked Equations (SIMPLE): what is it?

Khawaja (Khawaja and Moatamedi 2018) wrote a clear explanation of this algorithm.

The continuous equations of looks like this:

\begin{aligned} & \frac{\partial u}{\partial t}=-\frac{u \partial u}{\partial x}-\frac{u \partial v}{\partial y}-\frac{\partial p}{\partial x}+\frac{\mu}{\rho}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right) \\ & \frac{\partial v}{\partial t}=-\frac{v \partial v}{\partial y}-\frac{v \partial u}{\partial x}-\frac{\partial p}{\partial y}+\frac{\mu}{\rho}\left(\frac{\partial^2 v}{\partial y^2}+\frac{\partial^2 v}{\partial x^2}\right) \end{aligned}

The first step is to solve the discretized momentum equations at the current time step:

\begin{gathered} u^{t+1}(i, j)=u^t(i, j)+ \\ \Delta t\left(\text { function }\left(u^t, v^t, \rho, \mu, \Delta x, \Delta y\right)-\frac{(p(i, j)-p(i, j+1))}{\Delta x}\right) \\ v^{t+1}(i, j)=v^t(i, j)+ \\ \Delta t\left(\text { function }\left(u^t, v^t, \rho, \mu, \Delta x, \Delta y\right)-\frac{(p(i, j)-p(i+1, j))}{\Delta y}\right) \end{gathered}

Then continuity equation in the discretized form:

\begin{equation} \frac{\left(u^{t+1}(i, j-1)-u^{t+1}(i, j)\right)}{\Delta x}+\frac{\left(v^{t+1}(i+1, j)-v^{t+1}(i, j)\right)}{\Delta y}=\text { residual } \end{equation}

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