Discontinuous Galerkin and Finite Difference Methods for the Acoustic Equations with Smooth Coefficients. Mario Bencomo TRIP Review Meeting 2013

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1 About Me Mario Bencomo Currently 2 nd year graduate student in CAAM department at Rice University. B.S. in Physics and Applied Mathematics (Dec. 2010). Undergraduate University: University of Texas at El Paso (UTEP). 1

2 Discontinuous Galerkin and Finite Difference Methods for the Acoustic Equations with Smooth Coefficients Mario Bencomo TRIP Review Meeting

3 Problem Statement Acoustic Equations (pressure-velocity form): ρ(x) v (x,t) + p(x,t) = 0 t (1a) 1 κ (x) p (x,t) + v(x,t) = f (x,t) t (1b) for x Ω and t [0,T ], where x = (x,z) and Ω = [0,1] 2. Boundary and initial conditions: p = 0, on Ω [0,T ] p(x,0) = p 0 (x) and v(x,0) = v 0 (x) Research focus: Analyze the computational efficiency of discontinuous Galerkin (DG) and finite difference (FD) methods in the context of the acoustic equations with smooth coefficients. 3

4 Problem Statement Acoustic Equations (pressure-velocity form): ρ(x) v (x,t) + p(x,t) = 0 t (1a) 1 κ (x) p (x,t) + v(x,t) = f (x,t) t (1b) for x Ω and t [0,T ], where x = (x,z) and Ω = [0,1] 2. Boundary and initial conditions: p = 0, on Ω [0,T ] p(x,0) = p 0 (x) and v(x,0) = v 0 (x) Research focus: Analyze the computational efficiency of discontinuous Galerkin (DG) and finite difference (FD) methods in the context of the acoustic equations with smooth coefficients. 3

5 Why Smooth Coefficients? Relevant for seismic applications: smooth trends in real data! Comparison has not been done before! Previous work with discontinuous coefficients (Wang, 2009). efficiency of DG over 2-4 FD (dome inclusion) DG resolves discontinuity interface errors 4

6 Focus of Talk 1 Discontinuous Galerkin (DG) Method derivation of scheme basis functions reference element 2 Locally conforming DG (LCDG) Method triangulation reference element basis functions 3 Summary and Future Work 5

7 Discontinuous Galerkin (DG) Method Why DG? (Cockburn, 2006; Brezzi et al., 2004; Wilcox et al., 2010) explicit time-stepping, after inverting block diagonal matrix can handle irregular meshes and complex geometries hp-adaptivity Idea: Approximate solution by piecewise polynomials on partitioned domain. Example: 1D piecewise linear approximation 6

8 Discontinuous Galerkin (DG) Method Why DG? (Cockburn, 2006; Brezzi et al., 2004; Wilcox et al., 2010) explicit time-stepping, after inverting block diagonal matrix can handle irregular meshes and complex geometries hp-adaptivity Idea: Approximate solution by piecewise polynomials on partitioned domain. Example: 1D piecewise linear approximation 6

9 Derivation of DG Semi-Discrete Scheme Let: Then, τ T, for some triangulation T on Ω test function w C (Ω) ρ v x t + p x τ = 0 = ρ v x t τ w dx + p x w dx = 0 I.B.P. and replace p with numerical flux p in boundary integral, τ ρ v x t τ w dx p w x τ dx + p w n x dσ = 0 IBP = ρ v x τ t τ w dx + p x τ w dx + (p p)w n x dσ = 0 (2) 7

10 Derivation of DG Semi-Discrete Scheme Let: Then, τ T, for some triangulation T on Ω test function w C (Ω) ρ v x t + p x τ = 0 = ρ v x t τ w dx + p x w dx = 0 I.B.P. and replace p with numerical flux p in boundary integral, τ ρ v x t τ w dx p w x τ dx + p w n x dσ = 0 IBP = ρ v x τ t τ w dx + p x τ w dx + (p p)w n x dσ = 0 (2) 7

11 Nodal Basis Functions Finite dimensional space W h : W h = {w : w τ P N (τ), τ T } Nodal DG: Use nodal basis, i.e., where P N (τ) = span{l τ j (x)}n j=1 τ T, Lagrange polynomials l τ j (xτ i ) = δ ij for given nodal set {x τ i }N i=1 τ N = 1 2 (N + 1)(N + 2), a.k.a., degrees of freedom per triangular element 8

12 Nodal Basis Functions Finite dimensional space W h : W h = {w : w τ P N (τ), τ T } Nodal DG: Use nodal basis, i.e., where P N (τ) = span{l τ j (x)}n j=1 τ T, Lagrange polynomials l τ j (xτ i ) = δ ij for given nodal set {x τ i }N i=1 τ N = 1 2 (N + 1)(N + 2), a.k.a., degrees of freedom per triangular element 8

13 Nodal Sets Example of nodal sets {x τ j }N j=1 : (a) N = 1,N = 3 (b) N = 2,N = 6 (c) N = 3,N = 10 9

14 DG Semi-Discrete Scheme Find v x,p W h such that τ ρ v x t τ w dx p x τ w dx + (p p)w n x dσ = 0 for all w W h and τ T. Note: v x W h = v x (x,t) τ = N j=1 v x (x τ j,t)lτ j (x), where {v x (x τ j,t)}n j=1 are unknowns. Same for v z and p, and numerical fluxes v x,v z,p. 10

15 Nodal Coefficient Vectors R N R N+1 v x (t) := [v x (x τ 1,t),v x(x τ 2,t),...,v x(x τ N,t)] T v z (t) := p(t) := v m n (t) := [v n (x τ m 1,t),v n (x τ m 2,t),...,v n (x τ m N+1,t)] T p m (t) := (v m n ) (t) := [vn(x τ m 1,t),vn(x τ m 2,t),...,vn(x τ m N+1,t)] T (p m ) (t) := where v n (x τ m j,t) = n m x v x (x τ m j,t) + n m z v z (x τ m j,t), and similar for v n(x τ m j,t). 11

16 DG Semi-Discrete Scheme From ρ v x τ t τ w dx + p x τ w dx + (p p)w n x dσ = 0, to where: MR d dt v x(t) + S x p(t) + 3 m=1 mass matrix (M) ij := mass matrix (M m ) ij := n m x M m ((p m ) p m )(t) = 0, τ eτ m l τ i τ l τ i lτ j dx, l τ i lτ m j dσ, in R N N in R N (N+1) stiffness matrix (S x ) ij := l τ j x dx, in RN N ρ-matrix (R) ij := δ ij ρ(x τ j ), in RN N 12

17 DG Semi-Discrete Scheme From ρ v x τ t τ w dx + p x τ w dx + (p p)w n x dσ = 0, to where: MR d dt v x(t) + S x p(t) + 3 m=1 mass matrix (M) ij := mass matrix (M m ) ij := n m x M m ((p m ) p m )(t) = 0, τ eτ m l τ i τ l τ i lτ j dx, l τ i lτ m j dσ, in R N N in R N (N+1) stiffness matrix (S x ) ij := l τ j x dx, in RN N ρ-matrix (R) ij := δ ij ρ(x τ j ), in RN N 12

18 DG Semi-Discrete Scheme System of ODE s: R d dt v x(t) = D x p(t) + for each τ T, where 3 m=1 n m x L m ((p m ) p m )(t) D x = M 1 S x, L m = M 1 M m. Similar result for other equations: R d dt v z(t) = D z p(t) + 3 m=1 K 1 d dt p(t) = f(t) Dx v x (t) D z v z (t) where n m z L m ((p m ) p m )(t) 3 m=1 L m ((v n ) v n )(t) (K ) ij = δ ij κ(x τ j ), (f) i = f l τ i dx, D z = M 1 S z. τ 13

19 DG Semi-Discrete Scheme System of ODE s: R d dt v x(t) = D x p(t) + for each τ T, where 3 m=1 n m x L m ((p m ) p m )(t) D x = M 1 S x, L m = M 1 M m. Similar result for other equations: R d dt v z(t) = D z p(t) + 3 m=1 K 1 d dt p(t) = f(t) Dx v x (t) D z v z (t) where n m z L m ((p m ) p m )(t) 3 m=1 L m ((v n ) v n )(t) (K ) ij = δ ij κ(x τ j ), (f) i = f l τ i dx, D z = M 1 S z. τ 13

20 DG Semi-Discrete Scheme Acoustic equations: DG scheme: R d dt v x(t) = D x p(t) + R d dt v z(t) = D z p(t) + ρ t v x = x p ρ t v z = z p 1 κ t p = f v 3 m=1 3 m=1 K 1 d dt p(t) = f(t) Dx v x (t) D z v z (t) n m x L m ((p m ) p m )(t) n m z L m ((p m ) p m )(t) 3 m=1 L m ((v n ) v n )(t) 14

21 Visualizing DG After time discretization (leapfrog): v n+1/2 x = v n 1/2 x tr [D 1 x p n + 3 m=1 n m x L m ((p m ) p m ) n ] t n+1/2 t n t n 1/2 15

22 Visualizing DG After time discretization (leapfrog): v n+1/2 x = v n 1/2 x tr 1 [D x p n + 3 m=1 n m x L m ((p m ) p m ) n ] t n+1/2 t n t n 1/2 16

23 Visualizing DG After time discretization (leapfrog): v n+1/2 x = v n 1/2 x tr 1 [D x p n + 3 m=1 n m x L m ((p m ) p m ) n ] t n+1/2 t n t n 1/2 17

24 Visualizing DG After time discretization (leapfrog): v n+1/2 x = v n 1/2 x tr 1 [D x p n + 3 m=1 n m x L m ((p m ) p m ) n ] t n+1/2 t n t n 1/2 18

25 Use of Reference Element in DG Reference triangle ˆτ: (-1,1) s r (-1,-1) (1,-1) Idea: Carry computations in ˆτ construct nodal set and basis functions in ˆτ: τ {l j } N j=1 s.t. l j(r i ) = δ ij for {r i } N i=1 ˆτ mass matrix computations l τ j lτ i dx = J(τ) l j l i dr = M = J(τ) ˆM ˆτ 19

26 Locally Conforming DG (LCDG) Method for Acoustics Why LCDG method? (Chung & Engquist, 2006, 2009) locally and globally energy conservative optimal convergence rate explicit time-step My contribution: nodal DG implementation of LCDG basis functions and nodal sets use of reference element 20

27 Locally Conforming DG (LCDG) Method for Acoustics Why LCDG method? (Chung & Engquist, 2006, 2009) locally and globally energy conservative optimal convergence rate explicit time-step My contribution: nodal DG implementation of LCDG basis functions and nodal sets use of reference element 20

28 Locally Conforming DG (LCDG) Method for Acoustics LCDG Idea: Enforce continuity of the pressure field and the normal component of the velocity field in a staggered manner. Example: 1D piecewise linear polynomial approximation 21

29 Triangulation Algorithm: unif_tri 1 given N sub, subdivide Ω into N sub N sub partition of squares 2 triangulate each square by adding a diagonal from top-left to bottom-right 3 Pick interior point at each triangle and re-triangulate Figure: Example of unif_tri for N sub = 2. 22

30 Triangulation Algorithm: unif_tri 1 given N sub, subdivide Ω into N sub N sub partition of squares 2 triangulate each square by adding a diagonal from top-left to bottom-right 3 Pick interior point at each triangle and re-triangulate Figure: Example of unif_tri for N sub = 2. 22

31 Triangulation Algorithm: unif_tri 1 given N sub, subdivide Ω into N sub N sub partition of squares 2 triangulate each square by adding a diagonal from top-left to bottom-right 3 Pick interior point at each triangle and re-triangulate Figure: Example of unif_tri for N sub = 2. 22

32 Triangulation Algorithm: unif_tri 1 given N sub, subdivide Ω into N sub N sub partition of squares 2 triangulate each square by adding a diagonal from top-left to bottom-right 3 Pick interior point at each triangle and re-triangulate Figure: Example of unif_tri for N sub = 2. 22

33 Triangulation Algorithm: unif_tri 1 given N sub, subdivide Ω into N sub N sub partition of squares 2 triangulate each square by adding a diagonal from top-left to bottom-right 3 Pick interior point at each triangle and re-triangulate Figure: Example of unif_tri for N sub = 2. 22

34 LCDG Approximation Spaces Pressure space P h : q P h if q τ P N (τ) q continuous on dashed edges q = 0 on Ω Velocity space V h : u V h if u τ P N (τ) 2 u n continuous dashed edges 23

35 LCDG Approximation Spaces Pressure space P h : q P h if q τ P N (τ) q continuous on dashed edges q = 0 on Ω Velocity space V h : u V h if u τ P N (τ) 2 u n continuous dashed edges 23

36 LCDG: Basis for P h Two types of pressure elements: boundary elements interior elements Reference elements: (-1,1) (-1,1) (1,1) s r (-1,-1) (1,-1) s r (-1,-1) (1,-1) 24

37 LCDG: Basis for P h Basis functions for boundary pressure elements (N = 1 example): 1 s Basis functions for pressure interior elements (N = 1 example): r 1 s 1 s 1 s 1 s r r r r 25

38 LCDG: Basis for V h Velocity elements and respective reference elements: (0, 2 3 3) velocity elements s r (0, 0) ( 1, 1 3 3) (1, 1 3 3) reference element 26

39 LCDG: Basis for V h Basis functions, N = 1 example: 27

40 LCDG Semi-Discrete Scheme For a triangulation T, find p P h and v V h such that Ω Ω ρ v t u dx B h (p,u) = 0 1 p κ t q dx + B h(v,q) = fq dx Ω for all q P h and u V h, where Bh (p,u) = p u dx + p [u ˆn] dσ Ω e Ep 0 e B h (v,q) = v q dx v ˆn[q] dσ. Ω e E e v 28

41 Summary standard DG motivation basis functions reference element LCDG triangulation spaces P h, V h reference elements and basis functions 29

42 Pending Work and Future Directions Pending work: implementation of LCDG absorbing BC (Chung & Engquist, 2009) time discretization (leapfrog and Runge-Kutta) error analysis Future directions: non-uniform triangulation 3D acoustics elasticity equations 30

43 References Brezzi, F., Marini, L., and Sïli, E. (2004). Discontinuous galerkin methods for first-order hyperbolic problems. Mathematical models and methods in applied sciences, 14(12): Chung, E. and Engquist, B. (2009). Optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions. SIAM Journal on Numerical Analysis, 47(5): Chung, E. T. and Engquist, B. (2006). Optimal discontinuous galerkin methods for wave propagation. SIAM Journal on Numerical Analysis, 44(5): Cockburn, B. (2003). Discontinuous Galerkin methods. ZAMM-Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik, 83(11): Wilcox, Lucas C., et al. A high-order discontinuous Galerkin method for wave propagation through coupled elasticðacoustic media. Journal of Computational Physics (2010): Hesthaven, J. S. and Warburton, T. (2007). Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, volume 54. Springer. Wang, X. (2009). Discontinuous Galerkin Time Domain Methods for Acoustics and Comparison with Finite Difference Time Domain Methods. Master s thesis, Rice University, Houston, Texas. Warburton, T. and Embree, M. (2006). The role of the penalty in the local discontinuous Galerkin method for Maxwell s eigenvalue problem. Computer methods in applied mechanics and engineering, 195(25):