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.