1. P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides, Strasbourg, France ackerer@imfs.u-strasbg.fr

2. P. Ackerer, IMFS, Barcelona 2006 2 OUTLINE 1. Introduction 2. Solving advective dominant transport 2.1. Eulerian methods: Finite Volumes, Finite Elements 3. Galerkin Discontinuous Finite Elements 3.1. 1D discretization 3.2. General formulation 3.3. Numerical integration 3.4. Slope limiter 4. Numerical experiments 4.1. 2D – 3D benchmarks 4.2. Comparisons with finite volumes 5. On going works

3. P. Ackerer, IMFS, Barcelona 2006 3 xjxj x j-1 x j-2 x j+1 x j-1/2 x j+1/2 Finite volumes xjxj x j-1 x j-2 x j+1 Finite elements xjxj x j-1 x j-2 x j+1 Discontinuous finite elements  x n+1 n n-1  t jj+1 j-1 j-2 Space/time discretization Introduction

4. P. Ackerer, IMFS, Barcelona 2006 4 Finite differences method (FD): Basic ideas: 1. Use Taylor’s (1685-1731) series 2. Replace the derivatives Richardson (1922) was first to apply FD to weather forecasting. It required 3 months' worth of calculations to predict weather for next 24 hours. Introduction

5. P. Ackerer, IMFS, Barcelona 2006 5 xjxj x j-1 x j-2 x j+1 x j-1/2 x j+1/2 u FV have a very strong physical meaning Finite Volumes methods Introduction

6. P. Ackerer, IMFS, Barcelona 2006 6 Some key numbers (1D) To reduce numerical diffusionTo avoid oscillation for this scheme (R. Courant, K. Friedrichs & H. Lewy,1924) Introduction

7. P. Ackerer, IMFS, Barcelona 2006 7 Galerkin Finite Elements method Basic ideas: 1. Approximate the unknown function by a sum of ‘simple’ functions withso that xjxj x j-1 x j-2 x j+1 FE 2. The numerical solution should be as close as possible to the exact solution over the domain for any with i=1 to ne,which leads to ne equations with ne unknowns xjxj x j-1 x j-2 x j+1 FV u Introduction

8. P. Ackerer, IMFS, Barcelona 2006 8 Basic ideas: 3. Choose which leads to 4. Standard Euler/implicit scheme for time discretization, for example written for i=1 to ne. The next steps are more or less easy mathematics... Introduction

9. P. Ackerer, IMFS, Barcelona 2006 9 Galerkin Discontinuous Finite elements method Basic ideas: 1. Approximate the unknown function by a sum of ‘simple’ functions INSIDE an element E xjxj x j-1 x j-2 x j+1 FE xjxj x j-1 x j-2 x j+1 DFE u Discontinuous Finite Elements 2. Defining on node/edge/face A inside of E and on edge/face A outside of E Y j (t) : degree of freedom (nodal conc., ….)

10. P. Ackerer, IMFS, Barcelona 2006 10 Basic ideas: 3. Second order explicit Runge-Kutta scheme : the flux through A, positive if pointed outside : norm of A (length, surface). Step 1: Step 2: Discontinuous Finite Elements

11. P. Ackerer, IMFS, Barcelona 2006 11 Basic ideas: 4. Oscillations avoided by slope limitation xjxj x j-1 x j-2 x j+1 Discontinuous Finite Elements

12. P. Ackerer, IMFS, Barcelona 2006 12 Hyperbolic 1D Variational form Linear approximation DGFE : 1D discretization

13. P. Ackerer, IMFS, Barcelona 2006 13 Galerkin formulation Discretization Explicit formulation leads to a local system: x i+1 xixi x i-1 x i+2 E DGFE : 1D discretization

14. P. Ackerer, IMFS, Barcelona 2006 14 x i+1 xixi x i-1 x i+2 E DGFE : 1D discretization Step 1: Step 2:

15. P. Ackerer, IMFS, Barcelona 2006 15 Slope limitation x i+1 xixi x i-1 x i+2 E DGFE : 1D discretization

16. P. Ackerer, IMFS, Barcelona 2006 16 General formulation Variational form : norm of A (length, surface). : the flux through A, positive if pointed outside Polynomial approximation Linear (2D): Bi-Linear (2D): DGFE : General formulation

17. P. Ackerer, IMFS, Barcelona 2006 17 Standard interpolation functions Bilinear interpolation Linear interpolation DGFE : General formulation

18. P. Ackerer, IMFS, Barcelona 2006 18 Step 1 : : the flux through A, positive if pointed outside : norm of A (length, surface). Step 2 : : depending on the sign of DGFE : General formulation

19. P. Ackerer, IMFS, Barcelona 2006 19 Numerical integration (1) Exact integration in reference element for E DGFE : Numerical integration

20. P. Ackerer, IMFS, Barcelona 2006 20 Exact numerical integration with Simpson’s rule (pol. Ordre 2) Numerical integration (2) DGFE : Numerical integration

21. P. Ackerer, IMFS, Barcelona 2006 21 DGFE : Slope limiting

22. P. Ackerer, IMFS, Barcelona 2006 22 Step 3 : Multidimensional slope limiter (Bilinear function) min(i)/max (i) : min/max of over each element containing i min(E)/max (E) : min/max value of over each element which has a common node with E. E Optimization : Constraints : then Extrema : DGFE : Slope limiting

23. P. Ackerer, IMFS, Barcelona 2006 23 1rd order Upwind Centered3rd order Upwind Implicit CFL=1 CFL=5 CFL=1 CFL=5 CFL=1 CFL=5 Crank- Nicholson CFL=1 CFL=5 CFL=1 CFL=5 CFL=1 CFL=5 1rd order BDF CFL=1 CFL=5 CFL=1 CFL=5 CFL=1 CFL=5 Flux discretisation Time discretization DGFE, CFL=1 FE, CFL=1 FE, CFL=5 DGFE : Numerical experiments 1D Benchmarks

24. P. Ackerer, IMFS, Barcelona 2006 24 Bilinear, CFL=0,6 Linear, CFL=0,6 Bilinear, CFL=0,1 DGFE : Numerical experiments 2D Benchmarks

25. P. Ackerer, IMFS, Barcelona 2006 25 Velocity field DGFE : Numerical experiments 3D Benchmarks

26. P. Ackerer, IMFS, Barcelona 2006 26 Finite volume Bilin. DGFE DGFE : Numerical experiments

27. P. Ackerer, IMFS, Barcelona 2006 27 Finite volume (CFL = 0.50) D-GFE (CFL = 0.50) EFD : 10000 cells, 30 000 unk. VF : 10000 cells, 10 000 unk., VF 2: 40000 cells, 40 000 unk. DGFE : Numerical experiments Comparisons with Finite Volumes

28. P. Ackerer, IMFS, Barcelona 2006 28 Discontinuous Galerkin: well known algorithms DGFE : Summary Efficient in tracking fronts Well adapted to change interpolation order from one element to the other BUT Explicit scheme …… Summary

29. P. Ackerer, IMFS, Barcelona 2006 29 DGFE : On going work Implicit upwind formulation : the flux through A, positive if pointed outside : norm of A (length, surface). Time domain decomposition

30. P. Ackerer, IMFS, Barcelona 2006 30 X t t+  t t+3  t/4 t+  t/2 t+  t/4 Time domain decomposition DGFE : On going work DGFE, CFL=1

31. P. Ackerer, IMFS, Barcelona 2006 31 Implicit upwind formulation DGFE : On going work

32. P. Ackerer, IMFS, Barcelona 2006 32 DGFE : On going work

33. P. Ackerer, IMFS, Barcelona 2006 33 DGFE : On going work

34. P. Ackerer, IMFS, Barcelona 2006 34 Next to come …. DGFE : On going work

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