2. 1 Lecture 24: Flux Limiters

3. 2 Last Time… l Developed a set of limiter functions l Second order accurate

4. 3 This Time… l Examine physical rationale for limiter functions l Application to unstructured meshes

5. 4 Recall Higher-Order Scheme for  e l Consider finding face value using a second-order scheme with the gradient found at the upwind cell: l Recall: l What is the limiter function trying to do?

6. 5 Limiter Functions  =2r

7. 6 Physical Interpretation l The value of r can be thought of as the ratio of two gradients: l Limiter chooses gradient adaptively to avoid creating extrema Downwind cell gradient Upwind cell gradient ww

8. 7 Case (a): Linear  Variation l Since: l If variation is a straight line, on a uniform mesh, r=1 l From our limiter function range,  =1 for r=1 l Can use either gradient and get the right value at e r=1

9. 8 Case (b): 2>r>1 l r>1 means l If we used  =1, we would not create overshoot l In fact we can use  up to r and not create

10. 9 Case (b): 2>r>1 (Cont’d) l Consider case when r e >1, i.e., l Say we choose the  =r e line l When  =r e :

11. 10 Case (b’): r>2 l Consider case when r e >2, i.e., l For r e >2, say we choose the  =2 line l When  =2:

12. 11 Case (c): 0< r<1 l If r<1:

13. 12 Case (c): 0<r<1 (Cont’d) l Consider case when 0<r e <1, i.e., l Say we choose  =r e l When  =r e :

14. 13 Case (d): r<0 l When r<0, this implies local extremum l Our limiter has  =0 for r<0 l This implies Defaults to first order upwind scheme

15. 14 Unstructured Meshes l Find face value using: l No easy way to define r f

16. 15 Unstructured Meshes Create fictitious point U Find value at U by using cell gradient Hence define r f

17. 16 Closure In this lecture, we l Considered the physical meaning of the limiter function l Saw that it was an adaptive way to choose either an upwind or a downwind gradient to find face value l Looked at difficulties in implementing for unstructured meshes