# Session 10

#### Convection-Diffusion 1d

Consider the initial value problem of finding u(x,t) such that

• du/dt + v*du/dx – eps*d2u/dx2 =0   (1)
• u(x,0) given

where v(x,t) is a given convection velocity, and eps a small positive diffusion coefficient of size h in computations with mesh size h. Introduce characteristics x(t) as solutions to

• dx/dt = v(x,t) for t > 0,
• x(0) given.

Compute by the chain rule (assuming eps vanishingly small):

• d/dt u(x(t),t) =  du/dx * dx/dt + du/dt = du/dt + v*du/dx =0    (by (1))

to see that the solution u(x,t) to (1) is constant along characteristics x(t). If v(x,t) = constant then characteristics are straight lines and u(x,t) represents translation of the initial wave form as shown in Convection-Diffusion 1d where eps = 0.15*h acts to stabilise the computation by smoothing. Extend to Convection-Diffusion 2d.

#### Fluid Mechanics 1d

The basic equation of fluid mechanics in 1d is Burgers’ equation: Find a function u(x,t) representing velocity such that

• du/dt + u*du/dx – eps*d2u/dx2= 0 for t > 0      (2)
• u(x,0) given

where eps is a small diffusion coefficient. We interprete (1) as a non-linear convection-diffusion equation with convection velocity u as a form of (1) with v=u. Computing solutions we find decreasing initial data to develop steeping fronts which propagate. Test other initial data and build experience. Compare with Traffic Model.

#### Fluid Mechanics 2d

Start with Burgers’ equation in 2d: Find velocities u(x,y,t) and v(x,y,t) such that for t>0:

• du/dt+u*du/dx + v*du/dy -eps*Δu =0  (momentum conservation in x)
• dv/dt+u*dv/dx + v*dv/dy – eps*Δv =0 (momentum conservation in y)

with u(x,y,0) and v(x,y,0) given and eps ≈ h = mesh size. See connection to 2d convection-diffusion equation (with u corresponding to vx and v to vy).

Then proceed to Euler’s equations for incompressible flow in 2d.