Stability and Geometric Conservation Laws

Transcription

1 and and Geometric Conservation Laws Dipartimento di Matematica, Università di Pavia Advanced Computational Methods for FSI May 3-7, Ibiza, Spain

2 and Introduction to We aim at discretizing a parabolic a Several authors studied this problem and various approaches are possible

3 and Introduction to We aim at discretizing a parabolic a Several authors studied this problem and various approaches are possible Arbitrary Lagrangian Eulerian (ALE) method, see <Hughes Liu Zimmermann 81>, <Donea 83>, and this lecture...

4 and Introduction to We aim at discretizing a parabolic a Several authors studied this problem and various approaches are possible Arbitrary Lagrangian Eulerian (ALE) method, see <Hughes Liu Zimmermann 81>, <Donea 83>, and this lecture... Space time approach, see <Tezduyar et al.> and <Masud Hughes 97>

5 and Introduction to We aim at discretizing a parabolic a Several authors studied this problem and various approaches are possible Arbitrary Lagrangian Eulerian (ALE) method, see <Hughes Liu Zimmermann 81>, <Donea 83>, and this lecture... Space time approach, see <Tezduyar et al.> and <Masud Hughes 97> Fictitious method, see <Glowinski Pan Periaux 94>

6 and Introduction to We aim at discretizing a parabolic a Several authors studied this problem and various approaches are possible Arbitrary Lagrangian Eulerian (ALE) method, see <Hughes Liu Zimmermann 81>, <Donea 83>, and this lecture... Space time approach, see <Tezduyar et al.> and <Masud Hughes 97> Fictitious method, see <Glowinski Pan Periaux 94> Level set method, see <Osher et al.> and <Sethian et al.>

7 and Introduction to We aim at discretizing a parabolic a Several authors studied this problem and various approaches are possible Arbitrary Lagrangian Eulerian (ALE) method, see <Hughes Liu Zimmermann 81>, <Donea 83>, and this lecture... Space time approach, see <Tezduyar et al.> and <Masud Hughes 97> Fictitious method, see <Glowinski Pan Periaux 94> Level set method, see <Osher et al.> and <Sethian et al.> Immersed boundary method, see <Peskin 77> and the lecture by L. Gastaldi

8 and Introduction to ALE (cont ed) Usually, structure dynamics described in Lagrangian coordinates (i.e., considering a fixed reference ).

9 and Introduction to ALE (cont ed) Usually, structure dynamics described in Lagrangian coordinates (i.e., considering a fixed reference ). On the other hand, it is often more convenient to describe fluid motion in Eulerian coordinates (i.e., in the physical )

10 and Introduction to ALE (cont ed) Usually, structure dynamics described in Lagrangian coordinates (i.e., considering a fixed reference ). On the other hand, it is often more convenient to describe fluid motion in Eulerian coordinates (i.e., in the physical ) When considering fluid-structure interaction problems, and in the presence of small deformations, in addition to the methods of the previous slide one can also consider Lagrangian coordinates and, for instance, transpiration conditions, see <Fernández Le Tallec 03>

11 and Introduction to ALE (cont ed) In this lecture we shall consider a simple a and show how to model it with the Arbitrary Eulerian Lagrangian (ALE) method.

12 and Introduction to ALE (cont ed) In this lecture we shall consider a simple a and show how to model it with the Arbitrary Eulerian Lagrangian (ALE) method. We focus on the use of finite elements for the space and on the mathematical proof of stability of the fully discretized system

13 and Introduction to ALE (cont ed) In this lecture we shall consider a simple a and show how to model it with the Arbitrary Eulerian Lagrangian (ALE) method. We focus on the use of finite elements for the space and on the mathematical proof of stability of the fully discretized system We mainly refer to <Gastaldi 01> and <B. Gastaldi 04> We acknowledge also <Formaggia Nobile 99> which inspired our work

14 and A simplified model Consider advection diffusion equation in a. variable t (0, T ), space Ω t

15 and A simplified model Consider advection diffusion equation in a. variable t (0, T ), space Ω t u t µ u + x (βu) = f x Ω t, t (0, T ) u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T )

16 and A simplified model Consider advection diffusion equation in a. variable t (0, T ), space Ω t u t µ u + x (βu) = f x Ω t, t (0, T ) u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T ) Functional framework and parameters: µ > 0, β W 1, (Q T ), f L 2 (Q T ), u 0 H 1 0 (Ω 0 ) with Q T = {(x, t) : t (0, T ), x Ω t }

17 and Variational formulation Find u such that for each t (0, T ), u(, t) H 1 0 (Ω t) and u Ω t t v dx + µ x u x v dx + x (βu)v dx Ω t Ω t = fv dx v H0 1 (Ω t ) Ω t u = u 0 in Ω 0, for t = 0

18 and Basic references: <Hughes Zimmermann 81> <Donea 83> <Donea Giuliani 89> Arbitrary Lagrangian Eulerian (ALE) formulation

19 and Basic references: <Hughes Zimmermann 81> <Donea 83> <Donea Giuliani 89> Arbitrary Lagrangian Eulerian (ALE) formulation Introduce suitable mapping from reference Ω 0 to actual Ω t A t : Ω 0 Ω t A t C 0 (Ω 0 ), A 1 t C 0 (Ω t )

20 and Basic references: <Hughes Zimmermann 81> <Donea 83> <Donea Giuliani 89> Arbitrary Lagrangian Eulerian (ALE) formulation Introduce suitable mapping from reference Ω 0 to actual Ω t A t : Ω 0 Ω t A t C 0 (Ω 0 ), A 1 t C 0 (Ω t ) ξ Ω 0 x = x(ξ, t) = A t (ξ) Ω t ALE coordinate spatial coordinate

21 and ξ (cont ed) A t x Ω 0 Ω t

22 and (cont ed) How to switch between frame given in spatial coordinates and frame given in ALE coordinates Consider a function φ : Q T R and the corresponding function in the ALE frame ˆφ : Ω 0 (0, T ) R.

23 and (cont ed) How to switch between frame given in spatial coordinates and frame given in ALE coordinates Consider a function φ : Q T R and the corresponding function in the ALE frame ˆφ : Ω 0 (0, T ) R. Then we have the following relationships ˆφ : Ω 0 (0, T ) R ˆφ(ξ, t) = φ(a t (ξ), t) = φ(x, t)

24 and (cont ed) How to switch between frame given in spatial coordinates and frame given in ALE coordinates Consider a function φ : Q T R and the corresponding function in the ALE frame ˆφ : Ω 0 (0, T ) R. Then we have the following relationships ˆφ : Ω 0 (0, T ) R ˆφ(ξ, t) = φ(a t (ξ), t) = φ(x, t) φ : Ω t (0, T ) R φ(x, t) = ˆφ(A 1 t (x), t) = ˆφ(ξ, t)

25 and (cont ed) How to switch between frame given in spatial coordinates and frame given in ALE coordinates Consider a function φ : Q T R and the corresponding function in the ALE frame ˆφ : Ω 0 (0, T ) R. Then we have the following relationships ˆφ : Ω 0 (0, T ) R ˆφ(ξ, t) = φ(a t (ξ), t) = φ(x, t) φ : Ω t (0, T ) R φ(x, t) = ˆφ(A 1 t (x), t) = ˆφ(ξ, t) Definition of ALE time derivative (x = A t (ξ)) φ φ t : Ω t (0, T ) R ξ t (x, t) := ˆφ (ξ, t) ξ t

26 and (cont ed) How to switch between frame given in spatial coordinates and frame given in ALE coordinates Consider a function φ : Q T R and the corresponding function in the ALE frame ˆφ : Ω 0 (0, T ) R. Then we have the following relationships ˆφ : Ω 0 (0, T ) R ˆφ(ξ, t) = φ(a t (ξ), t) = φ(x, t) φ : Ω t (0, T ) R φ(x, t) = ˆφ(A 1 t (x), t) = ˆφ(ξ, t) Definition of ALE time derivative (x = A t (ξ)) φ φ t : Ω t (0, T ) R ξ t (x, t) := ˆφ (ξ, t) ξ t Analogous notation for time derivative in the spatial frame φ t : Ω t (0, T ) R x

27 and (cont ed) Jacobian of the ALE mappings J t = x ξ, J t = det(j t ) Assumption: J t κ > 0 uniformly in t

28 and (cont ed) Jacobian of the ALE mappings J t = x ξ, J t = det(j t ) Assumption: J t κ > 0 uniformly in t Domain velocity (x = A t (ξ)) w(x, t) = x t (x, t) ξ

29 and (cont ed) Jacobian of the ALE mappings J t = x ξ, J t = det(j t ) Assumption: J t κ > 0 uniformly in t Domain velocity (x = A t (ξ)) w(x, t) = x t (x, t) ξ Chain rule gives the change of variable in the time derivative (function φ smooth enough) φ t (x, t) = φ ξ t (x, t) + x x t x φ(x, t) ξ = φ t (x, t) + w(x, t) x φ(x, t) x

30 and (cont ed) Important isomorphism (see <Formaggia Nobile 99>) Proposition If Ω t = A t (Ω 0 ) is bounded and Lipschitz continuous and if {A t, A 1 t } W 1, for any t, then φ H 1 (Ω t ) ˆφ H 1 (Ω 0 ) Moreover, φ H 1 (Ω t) is equivalent to ˆφ H 1 (Ω 0 )

31 and ALE variational formulation u t µ u + x (βu) = f x Ω t, t (0, T ) u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T )

32 and ALE variational formulation u t µ u + x (βu) = f x Ω t, t (0, T ) u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T ) Substitute time derivative with ALE time derivative and take into account change of variable u t µ u + x (βu) w x u = f x Ω t, t (0, T ) ξ u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T )

33 and ALE variational formulation u t µ u + x (βu) = f x Ω t, t (0, T ) u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T ) Substitute time derivative with ALE time derivative and take into account change of variable u t µ u + x (βu) w x u = f x Ω t, t (0, T ) ξ u = u 0 x Ω 0, t = 0 u = 0 x Ω t, t (0, T ) Functional space compatible with ALE mapping H(Ω t ) = {v : Ω t R : v = ˆv A 1 t, ˆv H 1 0 (Ω 0 ) t (0, T )}

34 and ALE variat. formul. (cont ed) Recall Reynolds transport formula ( ) d φ φ(x, t) dx = dt V t V t t + φ x w dx ξ ( ) φ = t + x φ w + φ x w dx x V t where V t Ω t is such that V t = A t (V 0 ) with V 0 Ω 0.

35 and ALE variat. formul. (cont ed) Recall Reynolds transport formula ( ) d φ φ(x, t) dx = dt V t V t t + φ x w dx ξ ( ) φ = t + x φ w + φ x w dx x V t where V t Ω t is such that V t = A t (V 0 ) with V 0 Ω 0. In particular, if v does not depend upon t, one has d v dx = v x w dx dt Ω t Ω t ( ) d φ φv dx = dt Ω t t + φ x w v dx ξ Ω t

36 and ALE variat. formul. (cont ed) Multiply equation by test function v and integrate by parts ( ) u Ω t t v + µ x u x v + x (βu)v w x uv dx ξ = fv dx Ω t

37 and ALE variat. formul. (cont ed) Multiply equation by test function v and integrate by parts ( ) u Ω t t v + µ x u x v + x (βu)v w x uv dx ξ = fv dx Ω t Consider LHS and use Reynolds formula u t v dx = d uv dx u x wv dx ξ dt Ω t Ω t Ω t Use also x (uw) = w x u + u x w

38 and ALE variat. formul. (cont ed) Finally, the variational formulation reads: find u : Ω t (0, T ) R s. t. u(t) H(Ω t ) t (0, T ), and d dt (u(t), v) t + a t (u(t), v) + b t (u(t), v) = (f, v) t v H(Ω t ) where (u, v) t = uv dx Ω t a t (u, v) = (µ x u x v + x (βu)v) dx Ω t b t (u, v) = x (uw)v dx Ω t

39 and Energy estimates By standard tools (take v = u(t), integrate by parts, use Reynolds formula and Gronwall lemma) it is possible to prove t u(t) 2 L 2 (Ω +µ t) x u 2 L 2 (Ω ds t) 0 t u 0 2 L 2 (Ω 0 ) + C f 2 H 1 (Ω ds t) 0

40 and Energy estimates By standard tools (take v = u(t), integrate by parts, use Reynolds formula and Gronwall lemma) it is possible to prove t u(t) 2 L 2 (Ω +µ t) x u 2 L 2 (Ω ds t) 0 t u 0 2 L 2 (Ω 0 ) + C f 2 H 1 (Ω ds t) 0 So far so good but the choice of A t has not been defined yet

41 and Choice of A t Suppose the evolution of the boundary is known, then one can use harmonic extension of the boundary position, or consider the reference as an elastic body which is deformed according to the evolution of its boundary.

42 and Choice of A t Suppose the evolution of the boundary is known, then one can use harmonic extension of the boundary position, or consider the reference as an elastic body which is deformed according to the evolution of its boundary. The latter approach guarantees the following important properties (see <Gastaldi 01>): Ω t = A t (Ω 0 ) bounded and Lipschitz continuous for any t A t invertible and {A t, A 1 t } W 1,

43 and FE semi For simplicity, Ω t convex polygon for any t FE construction is based on reference spaces defined on Ω 0 and approximation of ALE mappings A t.

44 and FE semi For simplicity, Ω t convex polygon for any t FE construction is based on reference spaces defined on Ω 0 and approximation of ALE mappings A t. T h,0 (symplectic) mesh of Ω 0 : reference space V h (Ω 0 ) = {v h H 1 0 (Ω 0 ) : v h K P k (K), K T h,0 }

45 and FE semi For simplicity, Ω t convex polygon for any t FE construction is based on reference spaces defined on Ω 0 and approximation of ALE mappings A t. T h,0 (symplectic) mesh of Ω 0 : reference space V h (Ω 0 ) = {v h H 1 0 (Ω 0 ) : v h K P k (K), K T h,0 } Approximation of ALE mappings A t by piecewise linears N h x h (ξ, t) = A h,t (ξ) = x i (t) ˆϕ i (ξ) where x i (t) = A t (ξ i ) is the image of the i-th node of T h,0 and ˆϕ i is the i-th linear shape function. i=1

46 and FE semi For simplicity, Ω t convex polygon for any t FE construction is based on reference spaces defined on Ω 0 and approximation of ALE mappings A t. T h,0 (symplectic) mesh of Ω 0 : reference space V h (Ω 0 ) = {v h H 1 0 (Ω 0 ) : v h K P k (K), K T h,0 } Approximation of ALE mappings A t by piecewise linears N h x h (ξ, t) = A h,t (ξ) = x i (t) ˆϕ i (ξ) where x i (t) = A t (ξ i ) is the image of the i-th node of T h,0 and ˆϕ i is the i-th linear shape function. Then actual FE spaces are given by i=1 H h (Ω t ) = {v h : Ω t R s.t. v h = ˆv h A h,t, ˆv h V h (Ω 0 )}

47 and Ω 0 ξ i FE semi (cont ed) A t x i Ω t

48 and ξ i FE semi (cont ed) A t x i Ω 0 Assume the Jacobian of A h,t satisfies det(j h,t ) κ > 0 Ω t

49 and ξ i FE semi (cont ed) A t x i Ω 0 Assume the Jacobian of A h,t satisfies det(j h,t ) κ > 0 T h,t = A h,t (T h,0 ) K t = A h,t (K) for K T h,0 Ω t

50 and ξ i FE semi (cont ed) A t x i Ω 0 Assume the Jacobian of A h,t satisfies det(j h,t ) κ > 0 T h,t = A h,t (T h,0 ) K t = A h,t (K) for K T h,0 If approximating mappings A h,t are linear, then actual FE spaces read Ω t H h (Ω t ) = {v h H 1 0 (Ω t ) : v h K P k (K), K T h,t }

51 and FE semi (cont ed) find u h : Ω t (0, T ) R s. t. u h (t) H h (Ω t ) t (0, T ): d dt (u h(t), v) t +a t (u h (t), v)+b t (u h (t), v) = (f, v) t v H h (Ω t ) where (u, v) t = uv dx Ω t a t (u, v) = (µ x u x v + x (βu)v) dx Ω t b t (u, v) = x (uw)v dx Ω t

52 and As for the continuous case of semi t u h (t) 2 L 2 (Ω +µ t) x u h 2 L 2 (Ω ds t) 0 t u 0,h 2 L 2 (Ω 0 ) + C f H 1 (Ω t) ds 0

53 and As for the continuous case of semi t u h (t) 2 L 2 (Ω +µ t) x u h 2 L 2 (Ω ds t) 0 t u 0,h 2 L 2 (Ω 0 ) + C f H 1 (Ω t) ds If u is smooth enough, then error estimate (see <Nobile 01>) 1 2 u(t) u h(t) 2 L 2 (Ω t) + µ 4 t 0 0 x (u(s) u h (s)) 2 L 2 (Ω t) ds 1 2 u(0) u 0,h 2 L 2 (Ω t) + Ch 2k ( u(t) 2 H k (Ω t) + t 0 ( u t with constant C dependent on w W 2, 2 H k (Ω s) + u 2 H k+1 (Ω s) ) ds )

54 and of semi For more general geometries (nonpolygonal, approximation of boundary), in the case of linear elements see <Gastaldi 01> t ũ(t) u h (t) 2 L 2 (Ω h,t ) + µ x (ũ(s) u h (s)) 2 L 2 (Ω h,s ) ds error of the initial datum + approximation error of the velocity + interpolation errors 0 + consistency terms due to the approximation of the

55 and of semi For more general geometries (nonpolygonal, approximation of boundary), in the case of linear elements see <Gastaldi 01> t ũ(t) u h (t) 2 L 2 (Ω h,t ) + µ x (ũ(s) u h (s)) 2 L 2 (Ω h,s ) ds error of the initial datum + approximation error of the velocity + interpolation errors 0 + consistency terms due to the approximation of the All terms are O(h 2 ), see next slide...

56 and of semi Assume w h is bounded in W 1, (Q T ) t (0, T ) let Ω t be s.t. Ω t Ω t and Ω h,t Ω t for all h ũ, β, f, and w extensions to Ω t regular enough

57 and of semi Assume w h is bounded in W 1, (Q T ) t (0, T ) let Ω t be s.t. Ω t Ω t and Ω h,t Ω t for all h ũ, β, f, and w extensions to Ω t regular enough Then t ũ(t) u h (t) 2 L 2 (Ω h,t ) + µ x (ũ(s) u h (s)) 2 L 2 (Ω h,s ) ds t ũ(0) u h (0) 2 L 2 ( Ω 0 ) + C w w h 2 L 2 (Ω h,s ) ds 0 0 t ( (ũ u I 2 ) + C 0 t + x (ũ u I ) 2 L 2 (Ω h,s ) L 2 (Ω h,s ) ) + ũ u I 2 L 2 (Ω h,s ) ds t ( ) + C h 2 ũ 2 H 2 ( Ω + s) x ( βũ) 2 H 1 ( Ω + f 2 ds s) H 1 ( Ω s) 0

58 and Let us consider a constant time step. Our problem can be seen as a system of ordinary differential equations d Y = G(Y, t) t (0, T ) dt Y(0) = Y 0 with Y : (0, T ) R N h G : R N h (0, T ) R N h differentiable cont. and Lipschitz cont. w.r.t. Y

59 and (cont ed) Commonly used numerical methods include Y n+1 Y n = tg(y n+1, t n+1 ) IE ( Y Y n+1 Y n + Y n ) = tg, t 2 n+1/2 MP Y n+1 Y n = t ( G(Y n+1, t n+1 ) + G(Y n, t n ) ) 2 CN 3 2 Yn+1 2Y n Yn 1 = tg(y n+1, t n+1 ) BDF

60 and (cont ed) Commonly used numerical methods include Y n+1 Y n = tg(y n+1, t n+1 ) IE ( Y Y n+1 Y n + Y n ) = tg, t 2 n+1/2 MP Y n+1 Y n = t ( G(Y n+1, t n+1 ) + G(Y n, t n ) ) 2 CN 3 2 Yn+1 2Y n Yn 1 = tg(y n+1, t n+1 ) BDF First three methods obtained after integration from t n and t n+1 and use of appropriate quadrature rule. Last method obtained through differentiation formulæ

61 and (cont ed) More precisely, consider quadrature rule Q(F ) Then, IE, MP, CN read tn+1 t n F (t) dt Y n+1 Y n = Q(G)

62 and (cont ed) More precisely, consider quadrature rule Q(F ) Then, IE, MP, CN read tn+1 t n F (t) dt Y n+1 Y n = Q(G) On the other hand, the construction of BDF is the following. Take second order interpolant using the values of Y n 1, Y n, and Y n+1 ; compute the derivative at t n+1 and enforce that it is equal to the right hand side G

63 and (cont ed) The ALE mappings need to be approximated in time: for first three schemes we use first order (linear) approximation A h, t (ξ, t) = t t n t A h,tn+1 (ξ) + t n+1 t A h,tn (ξ) t

64 and (cont ed) The ALE mappings need to be approximated in time: for first three schemes we use first order (linear) approximation A h, t (ξ, t) = t t n t A h,tn+1 (ξ) + t n+1 t A h,tn (ξ) t Discrete velocity approximated accordingly w n+1 h, t (ξ) = A h,t n+1 (ξ) A h,tn (ξ), for t (t n, t n+1 ) t w h, t (x, t) = w n+1 h, t (ξ) A 1 h,t n+1 (x) turns out to be a piecewise constant function

65 and (cont ed) The ALE mappings need to be approximated in time: for first three schemes we use first order (linear) approximation A h, t (ξ, t) = t t n t A h,tn+1 (ξ) + t n+1 t A h,tn (ξ) t Discrete velocity approximated accordingly w n+1 h, t (ξ) = A h,t n+1 (ξ) A h,tn (ξ), for t (t n, t n+1 ) t w h, t (x, t) = w n+1 h, t (ξ) A 1 h,t n+1 (x) turns out to be a piecewise constant function Note: BDF formula requires second order approximation of ALE mappings (using three time values). Hence linear velocity representation

66 and (cont ed) Useful notation M t1 (φ(t 2 )) : Ω t1 R M t1 (φ(t 2 )) = φ(t 2 ) A h,t2 A 1 h,t 1

67 and (cont ed) Useful notation M t1 (φ(t 2 )) : Ω t1 R M t1 (φ(t 2 )) = φ(t 2 ) A h,t2 A 1 h,t 1 Full scheme (cases 1-3) for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + Q a (a t (u h, v h )) + Q b (b t (u h, v h ; w h, t )) u 0 h = u h,0, = Q a ((f (t), v h ) t ) v h H(Ω tn+1 ),

68 and analysis We say that the scheme is stable if there exist two real numbers α and β such that the following inequality holds for all v n h H(Ω t n ) with n = 0,..., N: 1 n+1 v 2 h 2 L 2 (Ω tn+1 ) 1 2 v h n 2 L 2 (Ω tn ) + tµ x (αm t n (v n+1 h ) + βm t n (vh n )) 2 L 2 (Ω tn ) + tq a (( x (β(αv n+1 h + βvh n n+1 )), αvh + βvh n ) t) (v n+1 h, αv n+1 h + βm tn+1 (vh n )) t n+1 (vh n, αm t n (v n+1 h ) + βvh n ) t n + Q a (a t (v h, αv n+1 h + βvh n )) + Q b(b t (v h, αv n+1 h + βvh n ; w h, t)) where t n is a properly chosen point in the interval [t n, t n+1 ]

69 and analysis (cont ed) As usual stability and consistency imply convergence (see <B. Gastaldi 04>) Theorem If a scheme is stable in the sense of the previous definition and if the errors in the quadrature rules Q a and Q b tend to zero as t goes to zero, then the fully discrete solution converges towards the continuous one Proof

70 and Geometric Conservation Laws We start by considering for finite volume formulations Many papers have been devoted to this subject. The term seems to show up for the first time in <Thomas Lombard 79> We report on the presented in <Guillard Fahrat 00>

71 and Geometric Conservation Laws We start by considering for finite volume formulations Many papers have been devoted to this subject. The term seems to show up for the first time in <Thomas Lombard 79> We report on the presented in <Guillard Fahrat 00> Let s consider the conservation law u t + x F(u) = 0 on a Ω t (same notation as before) with suitable initial and boundary conditions

72 and Let K t be a cell in the FV With arguments similar as for the parabolic equation, the ALE formulation of the conservation law on the cell K t reads d u dx + (F(u) wu) n ds = 0 dt K t K t

73 and Let K t be a cell in the FV With arguments similar as for the parabolic equation, the ALE formulation of the conservation law on the cell K t reads d u dx + (F(u) wu) n ds = 0 dt K t K t Let δ/δt be an approximation of order k of the time derivative and Φ(u, v, n, w) a numerical flux

74 and The numerical scheme reads FV (cont ed) δ δt K u K + e K ē Φ(u m e L, u m e R, n e, w e ) = 0 where ē, n e, and w e are averaged values of e(t) n e = 1 n(s, t) ds e(t) e(t) w e = w(s, t) ds e(t)

75 and for finite volumes As usual, numerical flux is assumed to be conservative Φ(u, v, n, w) = Φ(u, v, n, w)

76 and for finite volumes As usual, numerical flux is assumed to be conservative Φ(u, v, n, w) = Φ(u, v, n, w) and consistent with the conservation law Φ(u, u, n, w) = F(u) n w nu

77 and for finite volumes As usual, numerical flux is assumed to be conservative Φ(u, v, n, w) = Φ(u, v, n, w) and consistent with the conservation law Φ(u, u, n, w) = F(u) n w nu Remark The evolution of a constant initial condition u(x, 0) = u 0 in an infinite is the constant solution u(x, t) = u 0 for any t

78 and for finite volumes As usual, numerical flux is assumed to be conservative Φ(u, v, n, w) = Φ(u, v, n, w) and consistent with the conservation law Φ(u, u, n, w) = F(u) n w nu Remark The evolution of a constant initial condition u(x, 0) = u 0 in an infinite is the constant solution u(x, t) = u 0 for any t property states that the discrete scheme computes exactly the constant solution

79 and for finite volumes (cont ed) δ δt K u K + e K In our framework, reads u 0 δ δt K + F(u 0) e K ē Φ(u m e L, u m e R, n e, w e ) = 0 ē n e u 0 ē w e n e = 0 e K

80 and for finite volumes (cont ed) δ δt K u K + e K In our framework, reads u 0 δ δt K + F(u 0) e K ē Φ(u m e L, u m e R, n e, w e ) = 0 ē n e u 0 e K ē w e n e = 0 For commonly used schemes, e K ē n e = 0, so that reduces to δ δt K ē w e n e = 0 e K

81 and δ δt K e K First order schemes ē w e n e = 0 In the case of first order schemes, has the expression K n+1 K n = t ē w e n e e K

82 and δ δt K e K First order schemes ē w e n e = 0 In the case of first order schemes, has the expression K n+1 K n = t e K ē w e n e It turns out that can be achieved by setting ē = ñ e t with g e gravity center of e and tn+1 ñ e = w e = w(g e ) n e = 1 ē t ñe t n e(t) n ds dt

83 and and stability for FV Assumption A.1: the function A t is such that the faces of each cell are planar for any t

84 and and stability for FV Assumption A.1: the function A t is such that the faces of each cell are planar for any t Assumption A.2: the scheme is first order consistent in space e Φ(u el (t), u er (t), n e, w e ) = e K e (F(u(g e, t) w e u(g e, t)) n e + O( e h h 2 ) e K

85 and and stability for FV Assumption A.1: the function A t is such that the faces of each cell are planar for any t Assumption A.2: the scheme is first order consistent in space e Φ(u el (t), u er (t), n e, w e ) = e K e K e (F(u(g e, t) w e u(g e, t)) n e + O( e h h 2 ) The following result has been proved in <Guillard Fahrat 00> together with more general (higher order approximation) Theorem Suppose that the numerical flux Φ(u, v, n, w) is conservative, consistent, and verifies hypothesis A.2. If the mesh satisfies assumption A.1 and the property is fulfilled, then the resulting scheme is first order accurate.

86 and for finite elements Here we follow <B. Gastaldi 04> Let s recall our FE formulation with for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + Q a (a t (u h, v h )) + Q b (b t (u h, v h ; w h, t )) u 0 h = u h,0, = Q a ((f (t), v h ) t ) v h H(Ω tn+1 ), a t (u, v) = (µ x u x v + x (βu)v) dx Ω t b t (u, v) = x (uw)v dx Ω t

87 and for finite elements Here we follow <B. Gastaldi 04> Let s recall our FE formulation with for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + Q a (a t (u h, v h )) + Q b (b t (u h, v h ; w h, t )) u 0 h = u h,0, = Q a ((f (t), v h ) t ) v h H(Ω tn+1 ), a t (u, v) = (µ x u x v + x (βu)v) dx Ω t b t (u, v) = x (uw)v dx Ω t We are interested in the evolution of constant field u = u 0

88 and for finite elements (cont ed) Setting u = u 0 constant gives ϕ i (t n+1 ) dx ϕ i (t n ) dx Ω tn+1 Ω tn ) = Q b ϕ i (t) x w h (t) dx i = 1,, N h ( Ω t

89 and for finite elements (cont ed) Setting u = u 0 constant gives ϕ i (t n+1 ) dx ϕ i (t n ) dx Ω tn+1 Ω tn ) = Q b ϕ i (t) x w h (t) dx ( Ω t i = 1,, N h In the framework of finite elements, it can be interesting to look at the following formula as well ( ) d φ φv dx = dt Ω t t + φ x w v dx ξ Ω t

90 and for finite elements (cont ed) Setting u = u 0 constant gives ϕ i (t n+1 ) dx ϕ i (t n ) dx Ω tn+1 Ω tn ) = Q b ϕ i (t) x w h (t) dx ( Ω t i = 1,, N h In the framework of finite elements, it can be interesting to look at the following formula as well ( ) d φ φv dx = dt Ω t t + φ x w v dx ξ Ω t which gives a more natural condition (see <Nobile 01>) ϕ i (t n+1 )ϕ j (t n+1 ) dx ϕ i (t n )ϕ j (t n )) dx Ω tn+1 Ω tn ) = Q b ϕ i (t)ϕ j (t) x w h (t) dx i, j = 1,, N h ( Ω t

91 and for finite elements (cont ed) ϕ i (t n+1 )ϕ j (t n+1 ) dx ϕ i (t n )ϕ j (t n )) dx Ω tn+1 Ω tn ) = Q b ϕ i (t)ϕ j (t) x w h (t) dx i, j = 1,, N h ( Ω t The fulfillment of such property is related to the precision order of quadrature rule Q b, indeed we have ϕ i (t n+1 )ϕ j (t n+1 ) dx ϕ i (t n )ϕ j (t n )) dx Ω tn+1 Ω tn = ϕ i (t)ϕ j (t) x w h (t) dx Ω t

92 and for finite elements (cont ed) Example: IE scheme reads for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + ta tn+1 (u n+1 h, v h ) + tb tn+1 (u n+1 h, v h ; w h, t ) = t(f (t n+1 ), v h ) tn+1 v h H(Ω tn+1 ), u 0 h = u h,0.

93 and for finite elements (cont ed) Example: IE scheme reads for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + ta tn+1 (u n+1 h, v h ) + tb tn+1 (u n+1 h, v h ; w h, t ) = t(f (t n+1 ), v h ) tn+1 v h H(Ω tn+1 ), u 0 h = u h,0. Quadrature rules are Q a (F ) = Q b (F ) = tf (t n+1 ) so that is not satisfied (quadrature rule Q b exact for constant functions only).

94 and for finite elements (cont ed) Example: IE scheme reads for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + ta tn+1 (u n+1 h, v h ) + tb tn+1 (u n+1 h, v h ; w h, t ) = t(f (t n+1 ), v h ) tn+1 v h H(Ω tn+1 ), u 0 h = u h,0. Quadrature rules are Q a (F ) = Q b (F ) = tf (t n+1 ) so that is not satisfied (quadrature rule Q b exact for constant functions only). Nevertheless, we are able to prove stability for t small enough

95 and for finite elements (cont ed) Scheme IE satisfies the following inequality 1 2 n+1 vh 2 n v h n 2 n + tµ x v n+1 h 2 n t x β(v n+1 h ) 2 dx Ω tn+1 C t 2 x w h, t (v n+1 h ) 2 dx Ω tn+1 (v n+1 h, v n+1 h ) tn+1 (vh n, M t n (v n+1 h )) n + ta tn+1 (v n+1 h, v n+1 h ) + tb tn+1 (v n+1 h, v n+1 h ; w h, t ),

96 and for finite elements (cont ed) Final convergence for IE scheme ( t small enough) u n+1 h u(t n+1 ) 2 L 2 (Ω tn+1 ) + tµ n i=0 x (u i+1 h u(t i+1 )) 2 L 2 (Ω ti+1 ) C u h,0 u 0 2 L 2 (Ω 0 ) ( tn+1 ( + Ch 2k u(t) 2 H k (Ω + u t) 2 ) ) 0 t + H k (Ω u 2 H k+1 (Ω s) s) ds tn+1 ( + C( t) 2 u h L x + u h 2 0 t ξ 2 (Ω t) t ξ L 2 (Ω t) + x u h 2 L 2 (Ω + f ) 2 t) t + f ξ L 2 (Ω 2 L 2 (Ω t) dt t) n + C( t) 3 u h 2 x t ξ i=0 L 2 (Ω ti+1 )

97 and for finite elements (cont ed) Modified IE scheme to ensure condition, see <Formaggia Nobile 99> and <Gastaldi 01>

98 and for finite elements (cont ed) Modified IE scheme to ensure condition, see <Formaggia Nobile 99> and <Gastaldi 01> Idea: take Q a (F ) = Q b (F ) = tf (t n+1/2 ) (midpoint rule) for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + ta tn+1/2 (M tn+1/2 (u n+1 h ), M tn+1/2 (v h )) + tb tn+1/2 (M tn+1/2 (u n+1 h ), M tn+1/2 (v h ); w h, t ) = t(f (t n+1/2 ), M tn+1/2 (v h )) tn+1/2 v h H(Ω tn+1 ), u 0 h = u h,0

99 and for finite elements (cont ed) Modified IE scheme to ensure condition, see <Formaggia Nobile 99> and <Gastaldi 01> Idea: take Q a (F ) = Q b (F ) = tf (t n+1/2 ) (midpoint rule) for n = 1,, N find u n h H(Ω t n ) such that (u n+1 h, v h ) tn+1 (uh n, M t n (v h )) tn + ta tn+1/2 (M tn+1/2 (u n+1 h ), M tn+1/2 (v h )) + tb tn+1/2 (M tn+1/2 (u n+1 h ), M tn+1/2 (v h ); w h, t ) = t(f (t n+1/2 ), M tn+1/2 (v h )) tn+1/2 v h H(Ω tn+1 ), u 0 h = u h,0 This scheme turns out to be unconditionally stable

100 and for finite elements (cont ed) Summary on theoretical presented in <B. Gastaldi 04> Scheme property Order IE mie CN mcn mbdf NO YES YES YES YES t small with respect to the velocity unconditionally stable t t h 2 and t small with respect to the velocity t h 2 and t small with respect to the velocity t small with respect to the velocity t + h k t + h k t 2 + h k t 2 + h k t 2 + h k

101 and First numerical test Let s prescribe the following motion (linear dilation, no need to interpolate). Initial is unit square A t (ξ) = x(ξ, t) = (2 cos(20πt))ξ

102 and First numerical test Let s prescribe the following motion (linear dilation, no need to interpolate). Initial is unit square A t (ξ) = x(ξ, t) = (2 cos(20πt))ξ and consider the following problem u t.01 u = 0 u = 0 in Ω t on Ω t u 0 = 1600x(1 x)y(1 y) in Ω 0

103 and First numerical test Let s prescribe the following motion (linear dilation, no need to interpolate). Initial is unit square A t (ξ) = x(ξ, t) = (2 cos(20πt))ξ and consider the following problem u t.01 u = 0 u = 0 in Ω t on Ω t u 0 = 1600x(1 x)y(1 y) in Ω 0 Aim of the test: check whether discrete energy is decreasing

104 and t =.02 blue t =.01 red t =.001 green t =.0001 magenta IE CN mie mcn BDF mbdf

105 and More numerical tests Consider different motion (initial always unit square), t (0, π) ( ) A 1 t (ξ) = x 1 ξ + sin(t) (ξ, t) = rigid motion η + 1 cos(t) A 2 t (ξ) = x 2 (ξ, t) = (2 cos(πt))ξ dilation ( A 3 t (ξ) = x 3 (1 + t(t 1)η)ξ (ξ, t) = η ) deformation into a trapezoid

106 and More numerical tests Consider different motion (initial always unit square), t (0, π) ( ) A 1 t (ξ) = x 1 ξ + sin(t) (ξ, t) = rigid motion η + 1 cos(t) A 2 t (ξ) = x 2 (ξ, t) = (2 cos(πt))ξ dilation ( A 3 t (ξ) = x 3 (1 + t(t 1)η)ξ (ξ, t) = η ) deformation into a trapezoid Aim of the tests: check the convergence rates

107 and Our meshes

108 and Tests parameters We plot the L ((0, T ); L 2 (Ω t )) error versus 1/h (we are expecting second order convergence in space, since we are considering the L 2 norm)

109 and Tests parameters We plot the L ((0, T ); L 2 (Ω t )) error versus 1/h (we are expecting second order convergence in space, since we are considering the L 2 norm) Choice of t: t h 2 for IE (first order) t h for CN and BDF (second order) In the following pictures optimal means second order

110 and Rigid motion 0 IE CN BDF2 0 mie mcn mbdf

111 and Dialation (red: t h 2 ) 0 IE CN BDF2 0 mie mcn mbdf

112 and Deformation into trapezoid 0 IE CN BDF2 0 mie mcn mbdf

113 and Some movies t=1*dt t=1*dt t=1*dt Rigid motion Dialation Deform. into trapezoid

114 and Partial differential equations on a with finite elements described in detail analysis addressed Geometric conservation laws introduced for finite volumes and finite elements seem more natural condition for finite volumes than for finite elements Academic numerical tests confirm that and stability are not equivalent for finite elements

115 and Partial differential equations on a with finite elements described in detail analysis addressed Geometric conservation laws introduced for finite volumes and finite elements seem more natural condition for finite volumes than for finite elements Academic numerical tests confirm that and stability are not equivalent for finite elements

116 and Partial differential equations on a with finite elements described in detail analysis addressed Geometric conservation laws introduced for finite volumes and finite elements seem more natural condition for finite volumes than for finite elements Academic numerical tests confirm that and stability are not equivalent for finite elements

117 and Partial differential equations on a with finite elements described in detail analysis addressed Geometric conservation laws introduced for finite volumes and finite elements seem more natural condition for finite volumes than for finite elements Academic numerical tests confirm that and stability are not equivalent for finite elements

118 and Partial differential equations on a with finite elements described in detail analysis addressed Geometric conservation laws introduced for finite volumes and finite elements seem more natural condition for finite volumes than for finite elements Academic numerical tests confirm that and stability are not equivalent for finite elements

119 and Partial differential equations on a with finite elements described in detail analysis addressed Geometric conservation laws introduced for finite volumes and finite elements seem more natural condition for finite volumes than for finite elements Academic numerical tests confirm that and stability are not equivalent for finite elements

120 and CIME course on mixed methods You all are kindly invited to attend the CIME course on mixed finite elements: Cetraro (Italy), June 26 - July 1, 2006 Organizers: D. Boffi L. Gastaldi

121 and CIME course on mixed methods You all are kindly invited to attend the CIME course on mixed finite elements: Cetraro (Italy), June 26 - July 1, 2006 Our featured speakers are: L. Demkowicz R.G. Duran R.S. Falk R. Stenberg Organizers: D. Boffi L. Gastaldi

122 and CIME course on mixed methods You all are kindly invited to attend the CIME course on mixed finite elements: Cetraro (Italy), June 26 - July 1, 2006 Our featured speakers are: L. Demkowicz R.G. Duran R.S. Falk R. Stenberg Organizers: D. Boffi L. Gastaldi

123 and Take v n h = un h Convergence proof and, after integration by parts, we get 1 2 un+1 h 2 L 2 (Ω tn+1 ) 1 2 un h 2 L 2 (Ω tn ) + tµ x (αm t n (u n+1 h ) + βm t n (uh n )) 2 L 2 (Ω tn ( ) 1 + tq a Ω t 2 x (β)(αu n+1 h + βuh n )2 dx (u n+1 h, αu n+1 h + βm tn+1 (uh n )) t n+1 (uh n, αm t n (u n+1 h ) + βuh n ) t n + Q a (a t (uh n, αun+1 h + βuh n )) + Q b (b t (uh n, αun+1 h + βuh n ; w h, t)) = Q a (f (t), αu n+1 h + βuh n ) t)

124 and Convergence proof (cont ed) Use Cauchy Schwarz and discrete version of Gronwall s lemma to obtain a priori estimate 1 2 un+1 h 2 L 2 (Ω tn+1 ) n + tµ x (αm t i (u h,i+1 ) + βm t n (u h,i )) 2 L 2 (Ω ti ) i=0 1 n+1 2 u h,0 2 L 2 (Ω 0 ) + C f (t i ) H 1 (Ω ti ) i=1

125 and Convergence proof (cont ed) Integrate semidiscrete equation from t n and t n+1, and subtract from fully discrete scheme to get error equation (u n+1 h = u h (t n+1 ), v h ) tn+1 (u n h u h(t n ), M tn (v h )) tn + Q a (a t (u n h u h(t), v h )) + Q b (b t (u n h u h(t), v h ; w h, t )) tn+1 t n a t (u h (t), v h ) dt Q a (a t (u h (t), v h )) + + tn+1 t n b t (u h (t), v h ; w h (t)) dt Q b (b t (u h (t), v h ; w h, t )) tn+1 t n (f (t), v h ) t dt Q a ((f (t), v h ) t )

126 and Convergence proof (cont ed) Integrate semidiscrete equation from t n and t n+1, and subtract from fully discrete scheme to get error equation (u n+1 h = u h (t n+1 ), v h ) tn+1 (u n h u h(t n ), M tn (v h )) tn + Q a (a t (u n h u h(t), v h )) + Q b (b t (u n h u h(t), v h ; w h, t )) tn+1 t n a t (u h (t), v h ) dt Q a (a t (u h (t), v h )) + + tn+1 t n b t (u h (t), v h ; w h (t)) dt Q b (b t (u h (t), v h ; w h, t )) tn+1 t n (f (t), v h ) t dt Q a ((f (t), v h ) t ) Need for suitable test function v h.

127 and Convergence proof (cont ed) Take v h = αu n+1 h + βuh n with α and β from the stability definition and find that difference uh n u h(t n ) L 2 (Ω tn can be bounded in terms of the errors in the quadrature rules Q a (applied to a t and RHS) and Q b (applied to b t ) back