Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P RINCIPLE FOR HIGHER ORDER B ACK TO 1D FOR 1 MORE SECOND . � x i + 1 φ i +1 / 2 ∂ f ∂ x ( u h ) dx ∀ [ x i , x i + 1 ] , φ i + 1 / 2 ( u h ) = 1 x i i i + 1 φ T ( u h ) = φ + i + 1 / 2 ( u h ) + φ − φ − i + 1 / 2 ( u h ) φ + Distribution : 2 i +1 / 2 i +1 / 2 Distribution i + 1 / 2 ( u h ) = ± ˆ φ ± coeff.s : f i + 1 / 2 ∓ f ( u i ) i i + 1 φ + φ − i − / 2 i +1 / 2 Compute nodal values : 3 solve algebraic system φ − i + 1 / 2 + φ + i − 1 / 2 = 0 ∀ i i i − 1 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S AME IN MULTI D: REINTERPRETATION OF VF AS RD div f ( u ) = 0 k n + j ij � � i � F ( u i , u j , n + ij ) + F ( u i , u j , n − ij ) = 0 n − ij j ∈ V ( i ) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S AME IN MULTI D: REINTERPRETATION OF VF AS RD div f ( u ) = 0 k n − ik n + ij j � � � i F ( u i , u j , n + ij ) + F ( u i , u k , n − ik ) = 0 K ∋ i R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S AME IN MULTI D: REINTERPRETATION OF VF AS RD div f ( u ) = 0 k n − ik n + ij j � � F ( u i , u j , n + ij ) + F ( u i , u k , n − ik ) i K ∋ i �� � n + ij + n − − f ( u i ) · = 0 ik R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S AME IN MULTI D: REINTERPRETATION OF VF AS RD div f ( u ) = 0 k � Φ K i = 0 K ∋ i Φ i := F ( u i , u j , n + ij ) + F ( u i , u k , n − ik ) n jk G � � n + ij + n − n ki − f ( u i ) · ik = F ( u i , u j , n + ij ) + F ( u i , u k , n − ik ) i − f ( u i ) · n i n ij 2 j R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S AME IN MULTI D: REINTERPRETATION OF VF AS RD div f ( u ) = 0 3 Φ i := F ( u i , u j , n + ij ) + F ( u i , u k , n − ik ) − f ( u i ) · n i n jk G 2 � � n ki f ( u i ) · n i Φ i = − 2 i ∈ K i � f h · n d ∂ K 1 = n ij ∂ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual 2
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S TREAMLINE DIFFUSION METHOD A STANDARD REMARK � � � � � � � a ( u h , v h ) = − ∇ v h · f ( u h ) + h K ∇ u f ( u h ) ∇ v h τ K ∇ u f ( u h ) ∇ u h dx Ω K K � � v h ˆ + f n ( g , u h ) dS . e e ∈E R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S TREAMLINE DIFFUSION METHOD A STANDARD REMARK � � � � � � � � � a ( u h , v h ) = ∇ v h · f ( u h ) dx + h K ∇ u f ( u h ) ∇ v h τ K ∇ u f ( u h ) ∇ u h dx K K K � � v h (ˆ + f n ( g , u h ) − f ( u h ) · n ) dS . e e ∈E R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P RINCIPLE FOR HIGHER ORDER Φ T � ∀ K ∈ T h compute : φ K = f h ( u h ) · � n 1 ∂ K φ K ( u h ) = � φ K Distribution : 2 i i ∈ K Φ T Distribution φ K i ( u h ) = sub-residuals coeff.s : Compute nodal values : 3 solve algebraic system � φ K i ( u h ) = 0 , ∀ i ∈ T h K | i ∈ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P RINCIPLE FOR HIGHER ORDER � ∀ K ∈ T h compute : φ K = Φ T f h ( u h ) · � n 1 ∂ K φ K ( u h ) = � φ K Distribution : 2 i i ∈ K Distribution φ K i ( u h ) = sub-residuals coeff.s : Φ T Compute nodal values : 3 solve algebraic system � φ K i ( u h ) = 0 , ∀ i ∈ T h K | i ∈ K � � �� � u n + 1 = u n φ K ( u h ) n i − ω i , ∀ i ∈ T h i i K | i ∈ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives D ESIGN PROPERTIES S TRUCTURAL CONDITIONS , BASIC PROPERTIES Under which conditions on the φ K i s we get Correct weak solutions (if convergent with h ) Formal k th order of accuracy Monotonicity (discrete max principle) Convergence (with h , and with n !) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives D ESIGN PROPERTIES S TRUCTURAL CONDITIONS , BASIC PROPERTIES Under which conditions on the φ K i s we get Correct weak solutions (if convergent with h ) Formal k th order of accuracy Monotonicity (discrete max principle) Convergence (with h , and with n !) Notation: DOF: σ i or M i or simply i R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 1 : CONSERVATION C ONSERVATION PRINCIPLE If there is a f h , continuous approximation of f such that φ K = � � φ K f h · ˆ j = n j ∈ K ∂ K example: f h = f ( u h ) or Lagrange interp. of f ( u i ) or . . . B ASIC RELATION Scheme : for all dof i , � φ K i ( u h ) = 0 (1) K ∋ i � � � introduce φ Gal , K K ψ i div f ( u h ) dx = K ∇ ψ i · f ( u h ) dx − ∂ K ψ i f ( u h ) · ˆ = nd σ i multiply (1) by test function v evaluated at i � � � � � �� � φ K i ( u h ) � � v i φ K � v i φ Gal , K � φ K i − φ Gal , K � 0 = v i = i = + v i i i i K ∋ i K i ∈ K K i ∈ K i ∈ K � � � 1 �� ∇ v h · f h ( u h ) dx + � i − φ Gal , K � φ K = ( v i − v j ) i N K ! Ω K i , j ∈ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 2 : ACCURACY . u ex , h interpolant of exact sol. assumed smooth Truncation error � � � � E ( u ex , h ; v ) := φ K i ( u ex , h ) v i i ∈T h K | i ∈ K G UIDING PRINCIPLE I ≡ E Galerkin II � �� � � �� � � � � 1 E ( u ex , h ; v ) = ∇ v h · f h ( u ex , h ) + ( v i − v j )( φ K i − φ Gal )( u ex , h ) i N K ! K ∈T h i , j ∈ K Ω R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 2 : ACCURACY . u ex , h interpolant of exact sol. assumed smooth Truncation error � � � � E ( u ex , h ; v ) := φ K i ( u ex , h ) v i i ∈T h K | i ∈ K G UIDING PRINCIPLE I ≡ E Galerkin II � �� � � �� � � � � 1 E ( u ex , h ; v ) = ∇ v h · f h ( u ex , h ) + ( v i − v j )( φ K i − φ Gal )( u ex , h ) i N K ! K ∈T h i , j ∈ K Ω K EY REMARK & FINAL RESULT � � ⇒ φ Gal , K ( u ex , h ) = T ∇ ψ i · f h ( u ex , h ) dx − ∂ K ψ i f h ( u ex , h ) · ˆ div f ( w ) = 0 = nd σ = i O ( h k + d ) Truncation error : |E ( u ex , h ; v ) | ≤ C ′ ( T h , u ex ) �∇ v � ∞ h k + 1 | φ K i ( u ex , h ) | ≤ C ′′ ( T h , u ex ) h k + d = O ( h k + d ) if (in d-D) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives K EY REMARK A SSUMING A REGULAR ENOUGHT EXACT SOLUTION if f h ( u ex , h ) − f ( u ex ) = O ( h k + 1 ) , for example Lagrange interpolation. � � φ Gal , K ( u ex , h ) = ∇ ψ i · f h ( u ex , h ) dx − ψ i f h ( u ex , h ) · ˆ nd σ i K ∂ K � � � � � � f h ( u ex , h ) − f ( u ex ) f h ( u ex , h ) − f ( u ex · ˆ = ∇ ψ i · dx − ψ i nd σ K ∂ K = | K | × O ( h − 1 ) × O ( h k + 1 ) + | ∂ K | × O ( h 0 ) × O ( h k + 1 ) = O ( h k + d ) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 2 : ACCURACY L INEARITY (A CCURACY ) PRESERVING SCHEMES � Since φ K ( w h ) = ∂ K f h ( u h ) · ˆ ndl = O ( h k + d ) schemes for which φ K i = β K i φ K with β K i uniformly bounded distribution coeff.s are formally k + 1 th order accurate (for k + 1 th order spatial interpolation) H OWEVER : G ODUNOV ’ S THEOREM The β K i must depend on the solution : A scheme cannot be both high order accurate and linear for a linear problem. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 2 : ACCURACY L INEARITY (A CCURACY ) PRESERVING SCHEMES � Since φ K ( w h ) = ∂ K f h ( u h ) · ˆ ndl = O ( h k + d ) schemes for which φ K i = β K i φ K with β K i uniformly bounded distribution coeff.s are formally k + 1 th order accurate (for k + 1 th order spatial interpolation) H OWEVER : G ODUNOV ’ S THEOREM The β K i must depend on the solution : A scheme cannot be both high order accurate and linear for a linear problem. F UNDEMENTAL ASSUMPTION IN ALL THIS BUSINESS : � φ K i ( u h ) = 0 , ∀ i ∈ T h has a unique solution K | i ∈ K � �� � � i.e. u n + 1 = u n φ K ( u h ) n i − ω i , ∀ i ∈ T h must converges i i K | i ∈ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 3: P RESERVATION OF MONOTONY + ACCURACY S UMMARY G OAL Given any element K , a set of residuals { φ M i ( u h ) } i ∈ K , construct a set of residuals { φ H i ( u h ) } i ∈ K with φ H i ( u ex , h ) = O ( h k + d ) . S TRUIJS ’ “ LIMITER ” max ( 0 , φ M i /φ K ) β H i = � max ( 0 , φ M j /φ K ) j ∈ K H IGH ORDER RESIDUALS i ( u h ) } i ∈ K , � { φ M i ∈ K φ M i ( u h ) = φ K φ H i = β H i φ K . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives C ONDITION 3: P RESERVATION OF MONOTONY + ACCURACY S UMMARY G OAL Given any element K , a set of residuals { φ M i ( u h ) } i ∈ K , construct a set of residuals { φ H i ( u h ) } i ∈ K with φ H i ( u ex , h ) = O ( h k + d ) . S TRUIJS ’ “ LIMITER ” max ( 0 , φ M i /φ K ) β H i = � max ( 0 , φ M j /φ K ) j ∈ K H IGH ORDER RESIDUALS i ( u h ) } i ∈ K , � { φ M i ∈ K φ M i ( u h ) = φ K � � � � ∇ f u ( u h ) · ∇ u h � φ H i = β H i φ K + ε ( u h ) h K ∇ f u ( u h ) · ∇ ϕ i T dx K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives M OTIVATION FOR THIS TERM S OLVE ∂ u ∂ x = 0 ON [ 0 , 1 ] 2 1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 In both cases, φ K = 0 : these are steady solutions when φ H i = β K i φ K . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives M OTIVATION FOR THIS TERM S OLVE ∂ u ∂ x = 0 ON [ 0 , 1 ] 2 1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 In both cases, φ K = 0 : these are steady solutions when φ H i = β K i φ K . Cure : � � � � ∇ f u ( u h ) · ∇ u h � i φ K − φ H , K = β K → β K i φ K + h K ∇ f u ( u h ) · ∇ ϕ i T dx i K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A CLEARER PICTURE ? : 1D A LOCK AT THE ENTROPY PRODUCTION ∂ f ( u ) = 0 x ∈ [ 0 , 1 ] ∂ x u ( 0 ) = u 0 u ( 1 ) = u 1 . assume f ′ ( u 0 ) > 0 and f ′ ( u 1 ) < 0 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A CLEARER PICTURE ? : 1D A LOCK AT THE ENTROPY PRODUCTION ∂ f ( u ) = 0 x ∈ [ 0 , 1 ] ∂ x u ( 0 ) = u 0 u ( 1 ) = u 1 . assume f ′ ( u 0 ) > 0 and f ′ ( u 1 ) < 0 so that the solution is u = u 0 . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives 0 = x 0 < x 1 < . . . < x n − 1 , x n = 1. elements: K i + 1 / 2 = [ x i , x i + 1 ] . Assume P 1 . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives 0 = x 0 < x 1 < . . . < x n − 1 , x n = 1. elements: K i + 1 / 2 = [ x i , x i + 1 ] . Assume P 1 . The total residual: Φ K i + 1 / 2 = f ( u i + 1 ) − f ( u i ) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives 0 = x 0 < x 1 < . . . < x n − 1 , x n = 1. elements: K i + 1 / 2 = [ x i , x i + 1 ] . Assume P 1 . The total residual: Φ K i + 1 / 2 = f ( u i + 1 ) − f ( u i ) � � K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 − 1 Φ = β f ( u i + 1 − f ( u i ) . γ := β σ σ σ i + 1 2 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives 0 = x 0 < x 1 < . . . < x n − 1 , x n = 1. elements: K i + 1 / 2 = [ x i , x i + 1 ] . Assume P 1 . The total residual: Φ K i + 1 / 2 = f ( u i + 1 ) − f ( u i ) � � K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 − 1 Φ = β f ( u i + 1 − f ( u i ) . γ := β σ σ σ i + 1 2 2 u 2 : Entropy balance for U ( u ) = 1 � �� N − 1 � � � � K i − 1 / 2 K i + 1 / 2 E = u i β f ( u i + 1 ) − f ( u i ) + β f ( u i + 1 ) − f ( u i ) i i i = 0 � �� � 1 N − 1 � u h ∂ f K i + 1 / 2 K i + 1 / 2 ∂ x ( u h ) dx + = γ u i + γ u i + 1 / 2 f ( u i + 1 ) − f ( u i )] big ) i i + 1 0 i = 0 � 1 N − 1 � u h ∂ f K i + 1 / 2 ∂ x ( u h ) dx + = γ ( f ( u i + 1 ) − f ( u i ))( u i + 1 − u i ) . i + 1 0 i = 0 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives 0 = x 0 < x 1 < . . . < x n − 1 , x n = 1. elements: K i + 1 / 2 = [ x i , x i + 1 ] . Assume P 1 . The total residual: Φ K i + 1 / 2 = f ( u i + 1 ) − f ( u i ) � � K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 − 1 Φ = β f ( u i + 1 − f ( u i ) . γ := β σ σ σ i + 1 2 Entropy balance for U ( u ) = 1 2 u 2 : � 1 N − 1 � u h ∂ f K i + 1 / 2 ∂ x ( u h ) dx + E = γ ( f ( u i + 1 ) − f ( u i ))( u i + 1 − u i ) . i + 1 0 i = 0 K i + 1 / 2 Dissipative scheme: γ ( f ( u i + 1 ) − f ( u i ))( u i + 1 − u i ) ≥ 0 with a strict inequality i + 1 for at least one interval, i.e. f ( u i + 1 ) − f ( u i ) K i + 1 / 2 γ ≥ 0 . i + 1 u i + 1 − u i R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives 0 = x 0 < x 1 < . . . < x n − 1 , x n = 1. elements: K i + 1 / 2 = [ x i , x i + 1 ] . Assume P 1 . The total residual: Φ K i + 1 / 2 = f ( u i + 1 ) − f ( u i ) � � K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 K i + 1 / 2 − 1 Φ = β f ( u i + 1 − f ( u i ) . γ := β σ σ σ i + 1 2 Entropy balance for U ( u ) = 1 2 u 2 : � 1 N − 1 � u h ∂ f K i + 1 / 2 ∂ x ( u h ) dx + E = γ ( f ( u i + 1 ) − f ( u i ))( u i + 1 − u i ) . i + 1 0 i = 0 K i + 1 / 2 Dissipative scheme: γ ( f ( u i + 1 ) − f ( u i ))( u i + 1 − u i ) ≥ 0 with a strict inequality i + 1 for at least one interval, i.e. f ( u i + 1 ) − f ( u i ) K i + 1 / 2 γ ≥ 0 . i + 1 u i + 1 − u i K i + 1 / 2 : having an L ∞ stable scheme, only . . . The evaluation of β σ R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives � 1 N − 1 � u h ∂ f K i + 1 / 2 ∂ x ( u h ) dx + E = γ ( f ( u i + 1 ) − f ( u i ))( u i + 1 − u i ) . i + 1 0 i = 0 Adding the streamline term, i.e. � x i + 1 � ∂ f � 2 ∂ϕ σ � �� � � ∂ f θ ( u i + 1 − u i ) T ∂ x dx = ( u i + 1 − u i ) ϕ σ ( x i + 1 ) − ϕ σ ( x i )) ∂ u ∂ u x i will modify the entropy into � � 1 N − 1 � � � u h ∂ f f ( u i + 1 ) − f ( u i ) � ∂ f K i + 1 / 2 ∂ x ( u h ) dx + � )( u i + 1 − u i ) 2 E = γ + θ i + 1 u i + 1 − u i ∂ u 0 i = 0 � 1 0 u h ∂ f ∂ x ( u h ) dx provided that θ ≥ 1. and E ≤ R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives E XAMPLES OF MONOTONE SCHEMES M ONOTONE SCHEMES : THE R USANOV SCHEME (L OCAL L AX F RIEDRICHS ) Choice of Rusanov : not essential at all ! R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives E XAMPLES OF MONOTONE SCHEMES M ONOTONE SCHEMES : THE R USANOV SCHEME (L OCAL L AX F RIEDRICHS ) Centered linear first order distribution : � � � � � � = 1 K φ K + α � � φ Rv ∇ u f ( u h ) · ∇ ψ j ( u i − u j ) , α ≥ max � � i � � K j ∈ K j ∈ K K j � = i K number of DoF per element ψ j Lagrange basis fcn. relative to node j W HY THIS SCHEME ? The Rv scheme is cheap and has general formulation 1 The Rv scheme is monotone and energy stable in the P 1 case. 2 By far one of the most dissipative ones 3 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N UMERICAL EXAMPLE : ROTATION u v R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N UMERICAL EXAMPLE : ROTATION u v R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives G RID CONVERGENCE sum / TP 1 . pdfsum / TP 1 . pngsum / TP 1 . jpg sum / TP 2 . pdfsum ǫ L 2 ( P 1 ) h 1/25 0.50493E-02 1/50 0.14684E-02 1/75 0.74684E-03 1/100 0.41019E-03 O ls L 2 = 1.790 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N UMERICAL EXAMPLE : B URGER ’ S EQ . N � u 2 � 2 , u = 0 ∇ · 1 Shock u = − 0 . 5 u = 1 . 5 1 Expansion Fan 1 . 5 − 0 . 5 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N UMERICAL EXAMPLE : B URGER ’ S EQ . N LxF+PSI+Filter scheme, P 1 interpolation LxF+PSI+filter scheme, P 2 interpolation Shock captured in 1 or 2 cells R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives F INAL REMARK The scheme is L 2 stable: one can prove (linear case) error estimates similar to those of the stream-line diffusion method || u − u ex || graph norm ≤ Ch k + 1 / 2 . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives F INAL REMARK The scheme is L 2 stable: one can prove (linear case) error estimates similar to those of the stream-line diffusion method || u − u ex || graph norm ≤ Ch k + 1 / 2 . The scheme is no longer (formaly) L ∞ stable, in practice it does. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives F INAL REMARK The scheme is L 2 stable: one can prove (linear case) error estimates similar to those of the stream-line diffusion method || u − u ex || graph norm ≤ Ch k + 1 / 2 . The scheme is no longer (formaly) L ∞ stable, in practice it does. It is however possible to construct a refined analysis, and design better limiters so that one can have both L 2 + L ∞ , at least for triangles and second order accuracy. (work in progress) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHM The scheme consists in 4 steps : Evaluate the total residual, local (continuous interpolant) 1 Evaluate monotone residual (Rusanov) : local, 2 Evaluate high order residual : local 3 Gather residual : indirections, importance of good numering of the degrees of 4 freedom The scheme is local and easy to parallelise S OLUTION METHOD Jacobian free + LUSGS-ILU R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) � ∂ K f h · n f h interpolated by polynomial of degree p , Evaluation of total residual: exact quadrature: degre p only R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) � ∂ K f h · n f h interpolated by polynomial of degree p , Evaluation of total residual: exact quadrature: degre p only Evaluation of matrices β : done via local eigenstructure R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) � ∂ K f h · n f h interpolated by polynomial of degree p , Evaluation of total residual: exact quadrature: degre p only Evaluation of matrices β : done via local eigenstructure Evaluation of streamline term: � � ( ∇ u f ( u h ) ·∇ φ σ ) τ ( ∇ u f ( u h ) ·∇ u h ) ≈ | K | ω quad ( ∇ u f ( u h ) ·∇ φ )[ x q ] τ ( ∇ u f ( u h ) ·∇ u h )[ x q K quad quadratic form � � ( ∇ u f ( u h ) · ∇ u h )[ x q ] τ ( ∇ u f ( u h ) · ∇ u h )[ x q ] := Q u σ Stream ( σ ) = σ quad R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) � ∂ K f h · n f h interpolated by polynomial of degree p , Evaluation of total residual: exact quadrature: degre p only Evaluation of matrices β : done via local eigenstructure Evaluation of streamline term: � � ( ∇ u f ( u h ) ·∇ φ σ ) τ ( ∇ u f ( u h ) ·∇ u h ) ≈ | K | ω quad ( ∇ u f ( u h ) ·∇ φ )[ x q ] τ ( ∇ u f ( u h ) ·∇ u h )[ x q K quad quadratic form � � ( ∇ u f ( u h ) · ∇ u h )[ x q ] τ ( ∇ u f ( u h ) · ∇ u h )[ x q ] := Q u σ Stream ( σ ) = σ quad need to be disspiative : Q ≥ 0 and Q = 0 must imply ( ∇ u f ( u h ) · ∇ u h = 0 ω quad = 1 / # dofs , and much less quad points (take some of the Lagrange points only) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A LGORITHMIC COST loop on elements, no loop on faces (except boundary faces) Evaluation gradient: done before looping, small and local linear systems (precomputed and stored inverses) � ∂ K f h · n f h interpolated by polynomial of degree p , Evaluation of total residual: exact quadrature: degre p only Evaluation of matrices β : done via local eigenstructure Evaluation of streamline term: � � ( ∇ u f ( u h ) ·∇ φ σ ) τ ( ∇ u f ( u h ) ·∇ u h ) ≈ | K | ω quad ( ∇ u f ( u h ) ·∇ φ )[ x q ] τ ( ∇ u f ( u h ) ·∇ u h )[ x q K quad quadratic form � � ( ∇ u f ( u h ) · ∇ u h )[ x q ] τ ( ∇ u f ( u h ) · ∇ u h )[ x q ] := Q u σ Stream ( σ ) = σ quad need to be disspiative : Q ≥ 0 and Q = 0 must imply ( ∇ u f ( u h ) · ∇ u h = 0 ω quad = 1 / # dofs , and much less quad points (take some of the Lagrange points only) Indirections: evaluation of � K ∋ σ Φ K σ R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives O VERVIEW 1 M OTIVATING EXAMPLE 2 R ESIDUAL DISTRIBUTION SCHEMES (RDS) 3 T HE VISCOUS CASE 4 A PPLICATION TO NS EQUATIONS 5 C ONCLUSION -P ERSPECTIVES R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives RD WITH VISCOUS TERMS : WHAT ARE THE PROBLEMS ? � � div f a ( u ) − div ( K ( u ) . ∇ u ) = div f a ( u ) − K ( u ) . ∇ u = 0 Accuracy: coupling of convection and diffusion: one single operator Total residual: � � Φ K ( u h ) = f a ( u ) − K ( u h ) ∇ u h ) · n d ∂ K . ∂ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives RD WITH VISCOUS TERMS : WHAT ARE THE PROBLEMS ? � � div f a ( u ) − div ( K ( u ) . ∇ u ) = div f a ( u ) − K ( u ) . ∇ u = 0 Accuracy: coupling of convection and diffusion: one single operator Total residual: � � Φ K ( u h ) = f a ( u ) − K ( u h ) ∇ u h ) · n d ∂ K . ∂ K Major issue: ∇ u h not single valued on edges. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P OTENTIAL SOLUTIONS Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’) 1 � � f a ( u ) − K ( u h ) � Φ K ( u h ) = ∇ u h ) · n d ∂ K . ∂ K Loss of stencil compactness R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P OTENTIAL SOLUTIONS Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’) 1 Mixed type approach: H Nishikawa, NIA. 2 u and q = ∇ u seen as independant variables � � ∂ u ∂τ + div f a ( u ) − div K ( u ) q = 0 � � ∂ q ∂τ + 1 ∇ u − q = 0 T Hyperbolic system, explicit eigenvalues (0 is multiple eigenvalue) R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P OTENTIAL SOLUTIONS Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’) 1 Mixed type approach: H Nishikawa, NIA. 2 u and q = ∇ u seen as independant variables � � ∂ u ∂τ + div f a ( u ) − div K ( u ) q = 0 � � ∂ q ∂τ + 1 ∇ u − q = 0 T Hyperbolic system, explicit eigenvalues (0 is multiple eigenvalue) double the storage (for scalar), not completely clear for NS R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives P OTENTIAL SOLUTIONS Reconstruct the gradients using out-of-element information (Caraeni, circa 2000’) 1 Mixed type approach: H Nishikawa, NIA. 2 Gradient recovery and � ∇ u h globaly continuous approximation and 3 � � f a ( u ) − K ( u h ) � Φ K ( u h ) = ∇ u h ) · n d ∂ K . ∂ K R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives G RADIENT RECOVERY I DEA Obtained from super-convergent patch recovery introduced by O. C. Zienkiewicz and J. Z. Zhu, Int. J. Numer. Meth. Eng., 33, 1992 least square: the gradient are approximated with the same order as the unknowns. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives ZZ RECOVERY Assume that the numerical solution u h of the problem is known at each DOF of the grid to the k + 1-th order of accuracy. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives ZZ RECOVERY Assume that the numerical solution u h of the problem is known at each DOF of the grid to the k + 1-th order of accuracy. Aim : obtain the values of the solution gradient, � ∇ u h at all the DOFs with same order of accuracy . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives ZZ RECOVERY Assume that the numerical solution u h of the problem is known at each DOF of the grid to the k + 1-th order of accuracy. Aim : obtain the values of the solution gradient, � ∇ u h at all the DOFs with same order of accuracy . The components of the recovered gradient, at the generic DOF i , are written in a polynomial form as follows � � � � � � ∂ u h ∂ u h � � = p K a x = p K a y , � � and ∂ x � ∂ y � i i with p K ( x ) = ( 1 , x , y , x 2 , . . . , x k + 1 , x k y , . . . , y k + 1 ) , a x = ( a x 1 , a x 2 , . . . , a x m ) and a y = ( a y 1 , a y 2 , . . . , a y m ) . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives ZZ RECOVERY Assume that the numerical solution u h of the problem is known at each DOF of the grid to the k + 1-th order of accuracy. Aim : obtain the values of the solution gradient, � ∇ u h at all the DOFs with same order of accuracy . The components of the recovered gradient, at the generic DOF i , are written in a polynomial form as follows � � � � � � ∂ u h ∂ u h � � = p K a x = p K a y , � � and ∂ x � ∂ y � i i with p K ( x ) = ( 1 , x , y , x 2 , . . . , x k + 1 , x k y , . . . , y k + 1 ) , a x = ( a x 1 , a x 2 , . . . , a x m ) and a y = ( a y 1 , a y 2 , . . . , a y m ) . Assuming N s sampling points, ξ ℓ , ℓ = 1 . . . N i s , are available for each DOF i , minimize N i � ∂ u h � 2 N i � ∂ u h � 2 � � s s ∂ x ( ξ k ) − p K ∂ y ( ξ k ) − p K F x = F y = , and k a x k a y k = 1 k = 1 with p k = p ( ξ k ) . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives S AMPLING RECOVERY ◦ : the patch assembly points • : where the gradient is Q 1 P 1 recovered △ : super-convergent sampling points. Q 2 P 2 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives RD SCHEMES FOR VISCOUS PROBLEMS Linear scheme (Ni-LW) � � K ∇ u h �� i = Φ e � A · ∇ u h − ∇· Φ e + A ·∇ ψ i Ξ d Ω N e dof Ω e � � � ∇ u h − � + K ∇ ψ i · ∇ u h d Ω , Ω e Non-linear scheme α e � i = Φ e � 1 Φ e ˜ + u i − u j ) , N e N e dof dof j ∈ Σ e h j � = i Φ e ˆ i = β e i Φ e � � �� � ∇ u h �� � � A · ∇ u h − ∇· K � Φ e i = ˆ Φ e i + ε e h ( u h ) A ·∇ ψ i − ∇· K ∇ ψ i Ξ d Ω Ω e � � � ∇ u h − � + K ∇ ψ i · ∇ u h d Ω . Ω e R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives RD SCHEMES FOR VISCOUS PROBLEMS Linear scheme (Ni-LW) � � K ∇ u h �� i = Φ e � A · ∇ u h − ∇· Φ e + A ·∇ ψ i Ξ d Ω N e dof Ω e � � � ∇ u h − � + K ∇ ψ i · ∇ u h d Ω , Ω e Non-linear scheme α e � i = Φ e � 1 Φ e ˜ + u i − u j ) , N e N e dof dof j ∈ Σ e h j � = i Φ e ˆ i = β e i Φ e � � �� � ∇ u h �� � � A · ∇ u h − ∇· K � Φ e i = ˆ Φ e i + ε e h ( u h ) A ·∇ ψ i − ∇· K ∇ ψ i Ξ d Ω Ω e � � � ∇ u h − � + K ∇ ψ i · ∇ u h d Ω . Ω e R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives L INEAR ADVECTION - DIFFUSION EQUATION � � Ω = [ 0 , 1 ] 2 , a ·∇ u = ν div ∇ u , on the exact solution of the problem reads � � √ ξ 1 − 1 + 16 π 2 ν 2 , u = − cos ( 2 πη ) exp 2 ν with η = a y x − a x y and ξ = a x x + a y y . Here a = ( 0 , 1 ) K and ν = 0 . 01 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives L INEAR ADVECTION - DIFFUSION EQUATION � � Ω = [ 0 , 1 ] 2 , a ·∇ u = ν div ∇ u , on the exact solution of the problem reads � � √ ξ 1 − 1 + 16 π 2 ν 2 , u = − cos ( 2 πη ) exp 2 ν with η = a y x − a x y and ξ = a x x + a y y . Here a = ( 0 , 1 ) K and ν = 0 . 01 In this case Re ≈ 1, worst case scenario R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives L INEAR ADVECTION - DIFFUSION EQUATION Weighted area (1.95) Weighted area (1.01) Weighted area (2.19) -0.5 L2-Projection (2.07) -1 L2-Projection (1.45) L2-Projection (1.46) Least square (2.79) Least square (2.05) Least square (2.14) SPR-ZZ (2.80) SPR-ZZ (2.24) SPR-ZZ (3.06) -2 -1.5 -1 -2 -1.5 2 error) -3 2 error) 2 error) -2.5 -2 log(L log(L log(L -3 -4 -2.5 -3.5 -4 -3 -5 -4.5 -3.5 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -2.2 -2 -1.8 -1.6 -1.4 -1.2 log(h) log(h) log(h) L 2 error in the solution of the linear advection-diffusion problem on triangular girds with quadratic elements. Error of the solution (first column), error of the x -component of the gradient (second column) error of the y -component of the gradient (third column). R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives V ISCOUS B URGER EQUATION � u 2 � ∂ y = ν ∂ 2 u ∂ + ∂ u on Ω = [ 0 , 1 ] 2 , ∂ x 2 , ∂ x 2 with the following exact solution u = 2 νπ exp ( − ν y π 2 ) sin ( π x ) a + exp ( − ν y π 2 ) cos ( π x ) , with a > 1 . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives V ISCOUS B URGER EQUATION 0 Weighted area (2.13) Weighted area (1.94) Weighted area (1.24) -0.5 SPR-ZZ (2.16) SPR-ZZ (1.99) SPR-ZZ (1.14) -1 Weighted area (1.89) Weighted area (1.68) Weighted area (1.45) -0.5 SPR-ZZ (2.84) SPR-ZZ (2.26) SPR-ZZ (3.20) -1 -1 -2 -1.5 -2 -1.5 2 error) 2 error) 2 error) -3 -2.5 log(L log(L log(L -2 -3 -2.5 -4 -3.5 -3 -4 -5 -3.5 -4.5 -2 -1.5 -1 -2 -1.5 -1 -2 -1.5 -1 log(h) log(h) log(h) L 2 error in the solution of the viscous Burger problem on triangular girds with linear (dashed lines) and quadratic (solid lines) elements. Error of the solution (first column), error of the x -component of the gradient (second column) error of the y -component of the gradient (third column). R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A NISOTROPIC PURE DIFFUSION � � on Ω = [ 0 , 1 ] 2 , − div K ∇ u = 0 , with � 1 � 0 K = , 0 δ the problem has the following exact solution u = sin ( 2 π x ) e − 2 π y √ 1 /δ , and in the numerical simulations δ = 10 3 . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A NISOTROPIC DIFFUSION Weighted area (1.47) 1 Weighted area (1.47) 0 Weighted area (1.89) SPR-ZZ (1.93) SPR-ZZ (1.47) SPR-ZZ (1.65) Weighted area (1.08) Weighted area (1.17) Weighted area (2.36) -0.5 0.5 0 SPR-ZZ (2.73) SPR-ZZ (2.30) SPR-ZZ (2.54) -1 0 -1.5 -0.5 -1 -2 2 error) 2 error) 2 error) -1 -2.5 log(L log(L log(L -1.5 -2 -3 -2 -3.5 -3 -2.5 -4 -3 -4.5 -4 -3.5 -5 -2 -1.5 -1 -2 -1.5 -1 -2 -1.5 -1 log(h) log(h) log(h) L 2 error in the solution of the anisotropic diffusion problem on triangular girds with linear (dashed lines) and quadratic (solid lines) elements. Error of the solution (first column), error of the x -component of the gradient (second column) error of the y -component of the gradient (third column). R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives A NISOTROPIC DIFFUSION 0 Weighted area (2.13) Weighted area (2.07) Weighted area (2.12) 0.5 SPR-ZZ (1.93) SPR-ZZ (2.00) SPR-ZZ (2.13) -0.5 Weighted area (3.28) Weighted area (2.16) Weighted area (3.45) 0 SPR-ZZ (3.27) SPR-ZZ (3.16) SPR-ZZ (3.11) -1 -1 -0.5 -1.5 -1 -2 -1.5 -2 2 error) 2 error) 2 error) -2.5 -2 log(L log(L log(L -3 -2.5 -3 -3.5 -3 -4 -3.5 -4 -4.5 -4 -4.5 -5 -5 -5.5 -5 -2 -1.5 -1 -2 -1.5 -1 -2.5 -2 -1.5 -1 log(h) log(h) log(h) L 2 error in the solution of the anisotropic diffusion problem on uniform, structured grid of quadrangles, with linear (dashed lines) and quadratic (solid lines) elements. Error of the solution (first column), error of the x -component of the gradient (second column) error of the y -component of the gradient (third column). . R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives O VERVIEW 1 M OTIVATING EXAMPLE 2 R ESIDUAL DISTRIBUTION SCHEMES (RDS) 3 T HE VISCOUS CASE 4 A PPLICATION TO NS EQUATIONS 5 C ONCLUSION -P ERSPECTIVES R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives M ANUFACTURED SOLNS : ACCURACY TEST S OL MADE OF TRIGS FUNCTIONS 3 3.5 2.5 Observed Order Du Observed Order u 3 2 2.5 Observed order 1.5 2 2nd Order GG 2nd Order GG 2nd Order SPR-ZZ 2nd Order SPR-ZZ 3rd Order GG 3rd Order GG 3rd Order SPR-ZZ 3rd Order SPR-ZZ 1 1.5 Top: linear scheme; 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 Re Re Bottom: non linear scheme 3 3.5 Left: error on solution; 2.5 3 Observed Order Du Observed Order u Right: error on gradients 2 2.5 2 1.5 1.5 1 2nd Order GG 2nd Order GG 2nd Order SPR-ZZ 2nd Order SPR-ZZ 3rd Order GG 3rd Order GG 3rd Order SPR-ZZ 3rd Order SPR-ZZ 1 0 1 2 3 4 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 Re Re R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N ACA , M=0.5, R E =5000 R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N ACA , M=0.5, R E =5000 M M 0.55 0.55 0.5 0.5 0.45 0.45 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 Mach Number contours (top) and streamlines near the trailing edge (bottom) for the second (left) and third (right) order linear scheme. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N ACA , M=0.5, R E =5000 -0.5 -0.4 0 -0.3 Cp Cp 0.5 -0.2 P1 P1 1 P2 P2 -0.1 0 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 x/c x/c Pressure coefficient along the whole NACA-0012 airfoil for the second and third order simulations, with the same number of DOFs. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives N ACA , M=0.5, R E =5000 -0.002 -0.1 -0.001 -0.05 Cf Cf 0 0 0.05 0.001 0.1 P1 P1 P2 0.002 P2 0 0.2 0.4 0.6 0.8 1 0.7 0.75 0.8 0.85 0.9 0.95 x/c x/c Skin friction coefficient along the whole NACA-0012 airfoil for the second and third order simulations, with the same number of DOFs. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives L AMINAR FLOW AROUND A D ELTA WING M=0.5, R E =4000, A O A= 12 . 5 ◦ y A Λ x A A − A � � � � � � � � � � � � z σ x t c F IGURE : Left: Bottom and side views of the model of the delta wing: Λ = 75 ◦ , σ = 60 ◦ and t / c = 0 . 024. Right: a coarse mesh of tetrahedra use for the simulations. R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
Motivating example Residual distribution schemes (RDS) The viscous case Application to NS equations Conclusion-Perspectives L AMINAR FLOW AROUND A D ELTA WING M=0.5, R E =4000, A O A= 12 . 5 ◦ 0 10 ρ ρ u ρ v ρ w t ρ e 10 -2 -4 10 L2 Residual -6 10 -8 10 -10 10 0 500 1000 Iterations F IGURE : Left: Streamlines and slices of Mach number contours along and behind the delta wing, for a third order simulation on a fine grid. Right: Convergence history for the order sequencing (second and third order). R. Abgrall Non oscillatory FEM-like schemes for compressible fluid dynamics: the residual
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