Understanding Flux Limiters in Computational Fluid Dynamics

1. Lecture 24: Flux Limiters

2. Last Time… • Developed a set of limiter functions • Second order accurate

3. This Time… • Examine physical rationale for limiter functions • Application to unstructured meshes

4. Recall Higher-Order Scheme for e • Consider finding face value using a second-order scheme with the gradient found at the upwind cell: • Recall: • What is the limiter function trying to do?

5. =2r Limiter Functions

6. ww Physical Interpretation • The value of r can be thought of as the ratio of two gradients: • Limiter chooses gradient adaptively to avoid creating extrema Downwind cell gradient Upwind cell gradient

7. r=1 Case (a): Linear  Variation • Since: • If variation is a straight line, on a uniform mesh, r=1 • From our limiter function range,=1 for r=1 • Can use either gradient and get the right value at e

8. Case (b): 2>r>1 • r>1 means • If we used =1, we would not create overshoot • In fact we can use  up to r and not create

9. Case (b): 2>r>1 (Cont’d) • Consider case when re >1, i.e., • Say we choose the =re line • When =re :

10. Case (b’): r>2 • Consider case when re >2, i.e., • For re>2, say we choose the =2 line • When =2:

11. Case (c): 0< r<1 • If r<1:

12. Case (c): 0<r<1 (Cont’d) • Consider case when 0<re <1, i.e., • Say we choose =re • When =re :

13. Case (d): r<0 • When r<0, this implies local extremum • Our limiter has  =0 for r<0 • This implies Defaults to first order upwind scheme

14. Unstructured Meshes • Find face value using: • No easy way to define rf

15. Unstructured Meshes • Create fictitious point U • Find value at U by using cell gradient • Hence define rf

16. Closure In this lecture, we • Considered the physical meaning of the limiter function • Saw that it was an adaptive way to choose either an upwind or a downwind gradient to find face value • Looked at difficulties in implementing for unstructured meshes