Advection Diffusion Problems

NUMERICAL METHODS FOR DIFFERENTIAL EQNS

ADVECTION - DIFFUSION

given vector field

for our ease we go to 1D case

diffusive term   advection term

b.c.: u(0) = 0   u(L) = 1

We want to figure out an exact solution of the problem.

linear scl. when diffusive term dominates with respect to transport

when transport dominates with respect to

diffuse solution with 2 regions, 1 with strong variations,

the other almost constant high gradient regions = boundary layer

Dirichlet b.c.

could also have constant term

becomes global 1

global becomes

1st and last row adjusted accordingly to bc.

e.g. discretization of transport term:

(3rd line of the problem)

Classic 1D FE scheme 1D, equivalent to a centered

Laplace discretization     Central approximation finite difference scheme

PÉCLET NUMBER

-u_i-1 + 2u_i - u_i+1 +

exact solution?

Pe = _Bh/^2μ, similar meaning of Pe, tells relative importance of advection and diffusion terms

• U'' = 0 , S = 0 fundamental solution → substituting we get charact eqn a.b.c. → solution

1. + _{(1 - Re)/(4 - Pe)}
2. - _{(1 + Re)/(4 - Pe)}
3. - _{(4 - Pe)}
4. - _{(1 + Re)}

• number of nodes

Suppose for simplicity we even, Pe = 7 → Bh / 2μ (Adv dominates wrt Diff.)

• negative denominator
• solution oscillates!!!

Where is the problem?

• β_{uⁱ⁺¹ - u^i-1}/2
• is sensitive to the direction!!

• In the discretization we say something not meaningful from physical point of view!
• Still we have advantages: getting a2^nd order precision

We don’t want to change approximation, only to avoid oscillations work on Pe

Pe = _B(h)/^2/μ

• h < 2μ/β = 0.02

• It involves very small elements, big issue in 2D or 3D
• In practice increasing of number of elements not used

Other possibility: Choose something in order of convergence to avoid oscillation

• Upwind finite difference approximation
• only 1^st order accurate
• it adds artificial diffusion, "killing" oscillations
• β > 0

μ - (uⁱ⁺¹ + 2uⁱ - u^i-1)/h²

• + β_{uⁱ - u^i-1}/h
• = 0
• + β_{uⁱ⁺¹ - uⁱ}/h
• β < 0

1. - uⁱ - u^i-1 = uⁱ - u^i-1/h + u^i-1 - uⁱ/^1/2h + ...

= -uⁱ⁺¹/2h + 2uⁱ/2h + uⁱ - u^i-1/2h

It is, in practice, another diffusive term. Centered approximation of 1^st derivative

(μ + Bβ/2)

- u_i² + 2uⁱ - uuⁱ⁺¹ - u^i-1

= -

• Upwinding technique

equivalent to the original problem, in which physical diffusivity is increased of an artificial diffusion term, = _3Bh/²