2D Laplace Equation

Laplace equation is a 2D second order differential and appears in the Navier Stokes in the diffusion term; it also appears in the third equation of the 2D incompressible NS which is used to couple between pressure and velocity.
Actually in this post we will solve only the right hand side the equation:

∂2p∂x2+∂2p∂y2=0

We already seen how we discretize this - the central difference scheme:

pni+1,j−2pni,j+pni−1,jΔx2+pni,j+1−2pni,j+pni,j−1Δy2=0

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2D Burgers Equation

The Burgers' equation sums up the equations solved in the last two posts: the nonlinear convection and the diffusion; still being not very complicated, some analytical solutions exist and the numerical solution can be validated easily.
The set of equations to be solved is the following:

∂u∂t+u∂u∂x+v∂u∂y=ν(∂2u∂x2+∂2u∂y2)

∂v∂t+u∂v∂x+v∂v∂y=ν(∂2v∂x2+∂2v∂y2)

They all look familiar and to solve them we will be using the steps we already know.

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