Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction A reconstruction-enhanced discontinuous Galerkin method for hyperbolic problems V´ aclav Kuˇ cera Faculty of Mathematics and Physics Charles University Prague V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Discontinuous Galerkin method with reconstruction Finite volume method with reconstruction 1 Continuous Problem Space semidiscretization Discontinuous Galerkin method with reconstruction 2 Formulation Theoretical results and numerical experiments V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Finite volume method with reconstruction 1 Continuous Problem Space semidiscretization Discontinuous Galerkin method with reconstruction 2 Formulation Theoretical results and numerical experiments V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary ∂ Ω . Continuous Problem Find u : Q T = Ω × ( 0 , T ) → R such that ∂ u ∂ t + div f ( u ) = 0 in Q T , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , where f = ( f 1 , ··· , f d ) and f s , s = 1 ,..., d are Lipschitz- continuous fluxes in the direction x s , s = 1 ,..., d . V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Finite volume method with reconstruction 1 Continuous Problem Space semidiscretization Discontinuous Galerkin method with reconstruction 2 Formulation Theoretical results and numerical experiments V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles K ∈ T h . By F h we denote the set of all edges. For each Γ ∈ F h we define a unit normal vector n Γ . For each face Γ ∈ F I h there exist two neighbours K ( L ) Γ , K ( R ) ∈ T h . Γ 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 ) and Γ ∈ F I h we set v | ( L ) v | ( R ) = trace of v | K ( L ) on Γ , = trace of v | K ( R ) on Γ , Γ Γ Γ Γ V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles K ∈ T h . By F h we denote the set of all edges. For each Γ ∈ F h we define a unit normal vector n Γ . For each face Γ ∈ F I h there exist two neighbours K ( L ) Γ , K ( R ) ∈ T h . Γ 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 ) and Γ ∈ F I h we set v | ( L ) v | ( R ) = trace of v | K ( L ) on Γ , = trace of v | K ( R ) on Γ , Γ Γ Γ Γ V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles K ∈ T h . By F h we denote the set of all edges. For each Γ ∈ F h we define a unit normal vector n Γ . For each face Γ ∈ F I h there exist two neighbours K ( L ) Γ , K ( R ) ∈ T h . Γ 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 ) and Γ ∈ F I h we set v | ( L ) v | ( R ) = trace of v | K ( L ) on Γ , = trace of v | K ( R ) on Γ , Γ Γ Γ Γ V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Let T h be a partition of the closure Ω into a finite number of closed triangles K ∈ T h . By F h we denote the set of all edges. For each Γ ∈ F h we define a unit normal vector n Γ . For each face Γ ∈ F I h there exist two neighbours K ( L ) Γ , K ( R ) ∈ T h . Γ 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 ) and Γ ∈ F I h we set v | ( L ) v | ( R ) = trace of v | K ( L ) on Γ , = trace of v | K ( R ) on Γ , Γ Γ Γ Γ V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Definition We define the space of discontinuous piecewise polynomial functions S n h = { v ; v | K ∈ P n ( K ) ∀ K ∈ T h } , where P n ( K ) is the set of all polynomials on K of degree ≤ n . S 0 h - finite volume space, S n h - discontinuous Galerkin space, S N h , N > n - Higher order DG reconstructions. V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Definition We define the space of discontinuous piecewise polynomial functions S n h = { v ; v | K ∈ P n ( K ) ∀ K ∈ T h } , where P n ( K ) is the set of all polynomials on K of degree ≤ n . S 0 h - finite volume space, S n h - discontinuous Galerkin space, S N h , N > n - Higher order DG reconstructions. V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Definition We define the space of discontinuous piecewise polynomial functions S n h = { v ; v | K ∈ P n ( K ) ∀ K ∈ T h } , where P n ( K ) is the set of all polynomials on K of degree ≤ n . S 0 h - finite volume space, S n h - discontinuous Galerkin space, S N h , N > n - Higher order DG reconstructions. V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization Definition We define the space of discontinuous piecewise polynomial functions S n h = { v ; v | K ∈ P n ( K ) ∀ K ∈ T h } , where P n ( K ) is the set of all polynomials on K of degree ≤ n . S 0 h - finite volume space, S n h - discontinuous Galerkin space, S N h , N > n - Higher order DG reconstructions. V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization We integrate over K ∈ T h and apply Green’s theorem d � � K u ( t ) dx + ∂ K f ( u ) · n dS = 0 . dt We define u K ( t ) := 1 � ¯ K u ( t ) dx | K | and obtain d u K ( t )+ 1 � dt ¯ ∂ K f ( u ) · n dS = 0 . | K | We assume, that there exists a piecewise polynomial function U N h ( t ) ∈ S N h such that U N h ( x , t ) = u ( x , t )+ O ( h N + 1 ) , ∀ x ∈ Ω , ∀ t ∈ ( 0 , T ) . V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization We integrate over K ∈ T h and apply Green’s theorem d � � K u ( t ) dx + ∂ K f ( u ) · n dS = 0 . dt We define u K ( t ) := 1 � ¯ K u ( t ) dx | K | and obtain d u K ( t )+ 1 � dt ¯ ∂ K f ( u ) · n dS = 0 . | K | We assume, that there exists a piecewise polynomial function U N h ( t ) ∈ S N h such that U N h ( x , t ) = u ( x , t )+ O ( h N + 1 ) , ∀ x ∈ Ω , ∀ t ∈ ( 0 , T ) . V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Finite volume method with reconstruction Continuous Problem Discontinuous Galerkin method with reconstruction Space semidiscretization We integrate over K ∈ T h and apply Green’s theorem d � � K u ( t ) dx + ∂ K f ( u ) · n dS = 0 . dt We define u K ( t ) := 1 � ¯ K u ( t ) dx | K | and obtain d u K ( t )+ 1 � dt ¯ ∂ K f ( u ) · n dS = 0 . | K | We assume, that there exists a piecewise polynomial function U N h ( t ) ∈ S N h such that U N h ( x , t ) = u ( x , t )+ O ( h N + 1 ) , ∀ x ∈ Ω , ∀ t ∈ ( 0 , T ) . V´ aclav Kuˇ cera A reconstruction-enhanced discontinuous Galerkin method for
Recommend
More recommend
