Discontinuous Galerkin Method Space Semidiscretization Time Discretization The Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations Miloslav Feistauer, V´ aclav Kuˇ cera Faculty of Mathematics and Physics Charles University Prague Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Time Discretization Discontinuous Galerkin Method Space Semidiscretization 1 Continuous Problem Space semidiscretization Time Discretization 2 Semi-implicit Time Discretization Examples Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Discontinuous Galerkin Method Space Semidiscretization 1 Continuous Problem Space semidiscretization Time Discretization 2 Semi-implicit Time Discretization Examples Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let Ω ⊂ R 2 be a bounded domain with boundary ∂ Ω = Γ I ∪ Γ O ∪ Γ W . Continuous Problem Find w : Q T = Ω × ( 0 , T ) → R 4 such that ∂ R s ( w , ∇ w ) 2 2 ∂ w ∂ f s ( w ) ∑ ∑ ∂ t + = in Q T , ∂ x s ∂ x s s = 1 s = 1 where w = ( ρ , ρ v 1 , ρ v 2 , e ) T ∈ R 4 , f i ( w ) = ( ρ v i , ρ v 1 v i + δ 1 i p , ρ v 2 v i + δ 2 i p , ( e + p ) v i ) T , R i ( w , ∇ w ) = ( 0 , τ i 1 , τ i 2 , τ i 1 v 1 + τ i 2 v 2 + k ∂θ / ∂ x i ) T , � ∂ v i � + ∂ v j τ ij = λδ ij div v + 2 µ d ij ( v ) , d ij ( v ) = 1 . 2 ∂ x j ∂ x i Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let Ω ⊂ R 2 be a bounded domain with boundary ∂ Ω = Γ I ∪ Γ O ∪ Γ W . Continuous Problem Find w : Q T = Ω × ( 0 , T ) → R 4 such that ∂ R s ( w , ∇ w ) 2 2 ∂ w ∂ f s ( w ) ∑ ∑ ∂ t + = in Q T , ∂ x s ∂ x s s = 1 s = 1 where w = ( ρ , ρ v 1 , ρ v 2 , e ) T ∈ R 4 , f i ( w ) = ( ρ v i , ρ v 1 v i + δ 1 i p , ρ v 2 v i + δ 2 i p , ( e + p ) v i ) T , R i ( w , ∇ w ) = ( 0 , τ i 1 , τ i 2 , τ i 1 v 1 + τ i 2 v 2 + k ∂θ / ∂ x i ) T , � ∂ v i � + ∂ v j τ ij = λδ ij div v + 2 µ d ij ( v ) , d ij ( v ) = 1 . 2 ∂ x j ∂ x i Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let Ω ⊂ R 2 be a bounded domain with boundary ∂ Ω = Γ I ∪ Γ O ∪ Γ W . Continuous Problem Find w : Q T = Ω × ( 0 , T ) → R 4 such that ∂ R s ( w , ∇ w ) 2 2 ∂ w ∂ f s ( w ) ∑ ∑ ∂ t + = in Q T , ∂ x s ∂ x s s = 1 s = 1 where w = ( ρ , ρ v 1 , ρ v 2 , e ) T ∈ R 4 , f i ( w ) = ( ρ v i , ρ v 1 v i + δ 1 i p , ρ v 2 v i + δ 2 i p , ( e + p ) v i ) T , R i ( w , ∇ w ) = ( 0 , τ i 1 , τ i 2 , τ i 1 v 1 + τ i 2 v 2 + k ∂θ / ∂ x i ) T , � ∂ v i � + ∂ v j τ ij = λδ ij div v + 2 µ d ij ( v ) , d ij ( v ) = 1 . 2 ∂ x j ∂ x i Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization We add the thermodynamical relations � e � ρ − 1 p = ( γ − 1 )( e − ρ | v | 2 / 2 ) , 2 | v | 2 θ = / c v . and the following set of boundary conditions: Case Γ I : b) v | Γ I × ( 0 , T ) = v D = ( v D 1 , v D 2 ) T , a) ρ | Γ I × ( 0 , T ) = ρ D , � � 2 2 v j + k ∂θ ∑ ∑ on Γ I × ( 0 , T ) ; c) τ ij n i ∂ n = 0 j = 1 i = 1 b) ∂θ Case Γ W : on Γ W × ( 0 , T ) ; a) v Γ W × ( 0 , T ) = 0 , ∂ n = 0 2 b) ∂θ ∑ Case Γ O : on Γ O × ( 0 , T ) ; a) τ ij n i = 0 , j = 1 , 2 , ∂ n = 0 i = 1 Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Discontinuous Galerkin Method Space Semidiscretization 1 Continuous Problem Space semidiscretization Time Discretization 2 Semi-implicit Time Discretization Examples Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles: T h = { K i } i ∈ I . For two neighboring elements we set Γ ij = ∂ K i ∩ ∂ K j and for i ∈ I we define s ( i ) = { j ∈ I ; K j is a neighbour of K i } . By n ij we denote the unit outer normal to ∂ K i on the face Γ ij . Over T h we define the broken Sobolev space H k ( Ω , T h ) = { v ; v | K ∈ H k ( K ) ∀ K ∈ T h } and for v ∈ H 1 ( Ω , T h ) we set v | Γ ij = trace of v | K i on Γ ij , � v � Γ ij = 1 2 ( v | Γ ij + v | Γ ji ) , average of traces of v on Γ ij , [ v ] Γ ij = v | Γ ij − v | Γ ji , jump of traces of v on Γ ij , Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles: T h = { K i } i ∈ I . For two neighboring elements we set Γ ij = ∂ K i ∩ ∂ K j and for i ∈ I we define s ( i ) = { j ∈ I ; K j is a neighbour of K i } . By n ij we denote the unit outer normal to ∂ K i on the face Γ ij . Over T h we define the broken Sobolev space H k ( Ω , T h ) = { v ; v | K ∈ H k ( K ) ∀ K ∈ T h } and for v ∈ H 1 ( Ω , T h ) we set v | Γ ij = trace of v | K i on Γ ij , � v � Γ ij = 1 2 ( v | Γ ij + v | Γ ji ) , average of traces of v on Γ ij , [ v ] Γ ij = v | Γ ij − v | Γ ji , jump of traces of v on Γ ij , Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles: T h = { K i } i ∈ I . For two neighboring elements we set Γ ij = ∂ K i ∩ ∂ K j and for i ∈ I we define s ( i ) = { j ∈ I ; K j is a neighbour of K i } . By n ij we denote the unit outer normal to ∂ K i on the face Γ ij . Over T h we define the broken Sobolev space H k ( Ω , T h ) = { v ; v | K ∈ H k ( K ) ∀ K ∈ T h } and for v ∈ H 1 ( Ω , T h ) we set v | Γ ij = trace of v | K i on Γ ij , � v � Γ ij = 1 2 ( v | Γ ij + v | Γ ji ) , average of traces of v on Γ ij , [ v ] Γ ij = v | Γ ij − v | Γ ji , jump of traces of v on Γ ij , Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles: T h = { K i } i ∈ I . For two neighboring elements we set Γ ij = ∂ K i ∩ ∂ K j and for i ∈ I we define s ( i ) = { j ∈ I ; K j is a neighbour of K i } . By n ij we denote the unit outer normal to ∂ K i on the face Γ ij . Over T h we define the broken Sobolev space H k ( Ω , T h ) = { v ; v | K ∈ H k ( K ) ∀ K ∈ T h } and for v ∈ H 1 ( Ω , T h ) we set v | Γ ij = trace of v | K i on Γ ij , � v � Γ ij = 1 2 ( v | Γ ij + v | Γ ji ) , average of traces of v on Γ ij , [ v ] Γ ij = v | Γ ij − v | Γ ji , jump of traces of v on Γ ij , Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = { v ; v | K ∈ P p ( K ) ∀ K ∈ T h } , where P p ( K ) is the space of all polynomials on K of degree ≤ p . In order to derive a variational formulation, we multiply the ϕ ∈ H 2 ( Ω , T h ) , Navier-Stokes equations by a test function ϕ ϕ apply Green’s theorem on individual elements and other manipulations which take into account the discontinuity of the discrete functions between elements. Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
Discontinuous Galerkin Method Space Semidiscretization Continuous Problem Time Discretization Space semidiscretization We discretize the continuous problem in the space of discontinuous piecewise polynomial functions S h = { v ; v | K ∈ P p ( K ) ∀ K ∈ T h } , where P p ( K ) is the space of all polynomials on K of degree ≤ p . In order to derive a variational formulation, we multiply the ϕ ∈ H 2 ( Ω , T h ) , Navier-Stokes equations by a test function ϕ ϕ apply Green’s theorem on individual elements and other manipulations which take into account the discontinuity of the discrete functions between elements. Feistauer, Kuˇ cera The Discontinuous Galerkin Method for the Compressible...
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