Higher order solution of ODEs arising from DG space - PowerPoint PPT Presentation

Formulation of the problem Space discretization Time discretization Higher order solution of ODEs arising from DG space semi-discretization of nonstationary convection-diffusion problems Miloslav Vlas´ ak V´ ıt Dolejˇ s´ ı Charles University Prague Faculty of Mathematics and Physics Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Introduction Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Introduction Our aim : efficient numerical scheme for a simulation of unsteady compressible flows, Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Introduction Our aim : efficient numerical scheme for a simulation of unsteady compressible flows, model problem: scalar nonstationary nonlinear convection-diffusion equation, Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Introduction Our aim : efficient numerical scheme for a simulation of unsteady compressible flows, model problem: scalar nonstationary nonlinear convection-diffusion equation, space semi-discretization (e.g., DGFEM), Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Introduction Our aim : efficient numerical scheme for a simulation of unsteady compressible flows, model problem: scalar nonstationary nonlinear convection-diffusion equation, space semi-discretization (e.g., DGFEM), suitable time discretization (e.g. BDF) Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem R d be convex, Q T ≡ Ω × (0 , T ), Let Ω ⊂ I Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem R d be convex, Q T ≡ Ω × (0 , T ), we seek Let Ω ⊂ I u : Q T → I R such that Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem R d be convex, Q T ≡ Ω × (0 , T ), we seek Let Ω ⊂ I u : Q T → I R such that d ∂ u ∂ f s ( u ) ∑ ∂ t + = 휀 Δ u + g in Q T , ∂ x s s =1 Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem R d be convex, Q T ≡ Ω × (0 , T ), we seek Let Ω ⊂ I u : Q T → I R such that d ∂ u ∂ f s ( u ) ∑ ∂ t + = 휀 Δ u + g in Q T , ∂ x s s =1 Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem R d be convex, Q T ≡ Ω × (0 , T ), we seek Let Ω ⊂ I u : Q T → I R such that d ∂ u ∂ f s ( u ) ∑ ∂ t + = 휀 Δ u + g in Q T , ∂ x s s =1 Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Continuous problem R d be convex, Q T ≡ Ω × (0 , T ), we seek Let Ω ⊂ I u : Q T → I R such that d ∂ u ∂ f s ( u ) ∑ ∂ t + = 휀 Δ u + g in Q T , ∂ x s s =1 � u � ∂ Ω × (0 , T ) = u D , u ( x , 0) = u 0 ( x ) , x ∈ Ω , Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Space settings Let V ⊂ L 2 (Ω) be a space for exact solution Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Space settings Let V ⊂ L 2 (Ω) be a space for exact solution V h , p be a space for discrete solution Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Diffusive form A h A h ( v , w ) be linear and symmetric Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Diffusive form A h A h ( v , w ) be linear and symmetric ∣∣∣ v ∣∣∣ 2 = A h ( v , v ) ∀ v ∈ V Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Diffusive form A h A h ( v , w ) be linear and symmetric ∣∣∣ v ∣∣∣ 2 = A h ( v , v ) ∀ v ∈ V A h ( v , w ) ≤ C ∣∣∣ v ∣∣∣ ∣∣∣ w ∣∣∣ ∀ v , w ∈ V h , p Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Diffusive form A h A h ( v , w ) be linear and symmetric ∣∣∣ v ∣∣∣ 2 = A h ( v , v ) ∀ v ∈ V A h ( v , w ) ≤ C ∣∣∣ v ∣∣∣ ∣∣∣ w ∣∣∣ ∀ v , w ∈ V h , p A h ( v − R h v , w ) = 0 ∀ w ∈ V h , p Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Diffusive form A h A h ( v , w ) be linear and symmetric ∣∣∣ v ∣∣∣ 2 = A h ( v , v ) ∀ v ∈ V A h ( v , w ) ≤ C ∣∣∣ v ∣∣∣ ∣∣∣ w ∣∣∣ ∀ v , w ∈ V h , p A h ( v − R h v , w ) = 0 ∀ w ∈ V h , p ∥ R h v − v ∥ ≤ Ch p +1 Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Convective form b h b h ( v , w ) be nonlinear in v and linear in w Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Convective form b h b h ( v , w ) be nonlinear in v and linear in w b h ( u , w ) − b h ( v , w ) ≤ C ∥ u − v ∥ ∣∣∣ w ∣∣∣ ∀ u , v , w ∈ V Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Convective form b h b h ( v , w ) be nonlinear in v and linear in w b h ( u , w ) − b h ( v , w ) ≤ C ∥ u − v ∥ ∣∣∣ w ∣∣∣ ∀ u , v , w ∈ V b h ( v , w ) − b h ( R h v , w ) ≤ Ch p +1 ∣∣∣ w ∣∣∣ Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Source form ℓ h ℓ h ( v ) be linear Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Semi-discrete problem find u h ∈ C 1 ([0 , T ]; V h , p ) such that Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Semi-discrete problem find u h ∈ C 1 ([0 , T ]; V h , p ) such that ( ∂ u h ) ∂ t ( t ) , v + 휀 A h ( u h ( t ) , v ) + b h ( u h ( t ) , v ) = ℓ h ( v ) ( t ) ∀ v ∈ V h , p , ∀ t ∈ [0 , T ] , Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Semi-discrete problem find u h ∈ C 1 ([0 , T ]; V h , p ) such that ( ∂ u h ) ∂ t ( t ) , v + 휀 A h ( u h ( t ) , v ) + b h ( u h ( t ) , v ) = ℓ h ( v ) ( t ) ∀ v ∈ V h , p , ∀ t ∈ [0 , T ] , ( u h (0) , v ) = ( u 0 , v ) ∀ v ∈ V h , p Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Semi-discrete problem find u h ∈ C 1 ([0 , T ]; V h , p ) such that ( ∂ u h ) ∂ t ( t ) , v + 휀 A h ( u h ( t ) , v ) + b h ( u h ( t ) , v ) = ℓ h ( v ) ( t ) ∀ v ∈ V h , p , ∀ t ∈ [0 , T ] , ( u h (0) , v ) = ( u 0 , v ) ∀ v ∈ V h , p Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Time discretization Let t s = s 휏 s = 0 , . . . , r be a partition of [0 , T ] with a time step 휏 = T / r , Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Time discretization Let t s = s 휏 s = 0 , . . . , r be a partition of [0 , T ] with a time step 휏 = T / r , h ≈ U s ∈ V h , p for s = 0 , . . . , r let u h ( t s ) = u s Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Euler method Backward Euler method linearized by Forward Euler method Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Euler method Backward Euler method linearized by Forward Euler method ( U s +1 − U s , v ) + 휏휀 A h ( U s +1 , v ) + 휏 b h ( U s , v ) = 휏ℓ ( v ) Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Euler method Backward Euler method linearized by Forward Euler method ( U s +1 − U s , v ) + 휏휀 A h ( U s +1 , v ) + 휏 b h ( U s , v ) = 휏ℓ ( v ) Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Euler method Backward Euler method linearized by Forward Euler method ( U s +1 − U s , v ) + 휏휀 A h ( U s +1 , v ) + 휏 b h ( U s , v ) = 휏ℓ ( v ) Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Euler method Backward Euler method linearized by Forward Euler method ( U s +1 − U s , v ) + 휏휀 A h ( U s +1 , v ) + 휏 b h ( U s , v ) = 휏ℓ ( v ) error estimates derived by Dolejˇ s´ ı, Feistauer, Hozman (2007) Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization Euler method Backward Euler method linearized by Forward Euler method ( U s +1 − U s , v ) + 휏휀 A h ( U s +1 , v ) + 휏 b h ( U s , v ) = 휏ℓ ( v ) error estimates derived by Dolejˇ s´ ı, Feistauer, Hozman (2007) h ,휏, L ∞ ( L 2 ) = O ( h 2 p + 휏 2 ) , ∥ e ∥ 2 Vlas´ ak, Dolejˇ s´ ı

Formulation of the problem Space discretization Time discretization BDF (one–leg) Vlas´ ak, Dolejˇ s´ ı