Upwind differencing scheme for convection

Imagine you are standing on a windy beach and you have to throw a ball at a target. You have to throw the ball in a way that the wind doesn't take it away from the target. This throwing technique is called "upwind".

Similarly, when we solve equations related to fluid flow, we need to find the direction in which the fluid is moving (like the wind on the beach), so that we can calculate how the fluid will behave in the future. The technique we use to find this direction is called "upwind differencing scheme".

In upwind differencing scheme, we divide the space into small grids like a checkerboard. Each grid has a value of fluid flow that we need to calculate. We also need to know the velocity of the fluid flowing in and out of each grid.

Based on this velocity, we decide whether we use the upwind direction or the downwind direction to calculate the value of fluid flow. If the fluid is moving more towards the right, we use the upwind direction (like throwing the ball on a windy beach). But, if the fluid is moving more towards the left, we use the downwind direction.

This way, we can calculate the fluid flow values for each grid and predict how the fluid will behave in the future based on its current velocity.