The slope factorσ∗may be chosen by employing theMinModlimiter using σ

Numerical Simulation of Transport Processes in Porous Media WS 13/14

Dr. Olaf Ippisch, Ole Klein IWR, Universit¨at Heidelberg

Exercise Sheet No.9 Deadline: 14. January 2013

EXERCISE9 CONVECTIONDISPERSIONEQUATION WITHSLOPELIMITER

In this exercise you will implement a time explicit discretization scheme for the Convection-Dispersion equation

∂tc+∇ · −D˜ ∇c+c ~v

= 0 ~x∈Ω

c = f ~x∈∂Ω_D

~n· −D˜ ∇c+c ~v

= J_N ~x∈∂Ω_N

using slope limiters to reduce numerical diffusion. Again, you will implement a cell centered finite volume scheme (cf. last exercise):

In time stepkwe require for each grid cellE:

Z

E

c^k+1dV −τ Z

∂E\∂Ω_N

~

nE·D˜ ∇c^kdA+τ Z

∂E\∂Ω_N

~ nE ·~v

c^k_∗+ 1

2σ∗(h−τ|~nE·~v|)

dA= Z

E

c^kdV −τ Z

∂E∩∂Ω_N

JN dA

Herec^k_∗ denotes the average concentration on the cell adjacent to the face in upwind direction. Let us consider a faceein a two dimensional grid and assume elementEat the grid coordinates(i, j)to be the upwind element and elementFat the grid coordinates(i+ 1, j)to be the downwind element.

Then we havec^k_∗ :=c^k_E =c^k_(i,j).

The slope factorσ∗may be chosen by employing theMinModlimiter using

σ∗ = minmod

c^k_(i+1,j)−c^k_(i,j)

h , c^k_(i,j)−c^k_(i−1,j) h

,

with minmod(a, b) =







a if |a|<|b| ∧a·b >0, b if |a| ≥ |b| ∧a·b >0, 0 if a·b≤0,

The slope factors for faces with normal vectors iny-direction may be determined analogously.

A program which applies the slope limiter implementation to a simple groundwater test problem is implemented inmain.cc. Compare the output for the explicit case using slope limiters or just pure upwinding (You can use the SLOPE macro to switch between both cases). The interface of the assem- bler is given in the filesgodunov assembler.hhandgodunov utilities.hh. In the former of these files, you will have to implement the methodassemble explicit(...)such that the solver uses the slope limiter CD transport or not depending on the last argument.

5 Points