Numerical discretization of coupling conditions of hyperbolic conservation laws by high-order schemes Axel-Stefan Häck LuF Mathematik - Kontinuierliche Optimierung IGPM RWTH Aachen May 25th, 2014 Joint work with Prof. Michael Herty (RWTH Aachen) and Prof. Mapundi Banda (Stellenbosch University)
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Table of contents Model 1 Numerical Scheme 2 Coupling Condition in Scheme 3 Algorithm 4 Linear Case and Numerical Results 5 Left to do and Outlook 6 2 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Consider a model of (spatial) 1D flow on graphs. 3 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Consider a model of (spatial) 1D flow on graphs. A singe vertex with n adjacent arcs (which we extend to infinity). 3 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Consider a model of (spatial) 1D flow on graphs. A singe vertex with n adjacent arcs (which we extend to infinity). All arcs are parameterized by [ 0 , ∞ ) , such that the junction is located at x = 0 for all arcs. 3 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Consider a model of (spatial) 1D flow on graphs. A singe vertex with n adjacent arcs (which we extend to infinity). All arcs are parameterized by [ 0 , ∞ ) , such that the junction is located at x = 0 for all arcs. Has broad applications e.g. gas, traffic, and blood flow. 3 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Consider a model of (spatial) 1D flow on graphs. A singe vertex with n adjacent arcs (which we extend to infinity). All arcs are parameterized by [ 0 , ∞ ) , such that the junction is located at x = 0 for all arcs. Has broad applications e.g. gas, traffic, and blood flow. We assume the flux f ( · ) ∈ C 4 ( R 2 , R 2 ) and the u j ( t , x ) : R + 0 × R + 0 to be the conserved states on the arcs j = 1 , ..., n . 3 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Coupled PDEs Setting: Coupled PDEs ∂ t u j + ∂ x f ( u j ) = 0 , t ≥ 0 , x ≥ 0 , u j ( 0 , x ) = u j , o ( x ) , x ≥ 0 , Ψ( u 1 ( t , 0 +) , . . . , u n ( t , 0 +)) = 0 , t ≥ 0 , where Ψ : R 2 n → R n is the (possibly nonlinear) coupling condition (CC). With Ψ fulfilling the transversality condition we have existence and uniqueness see [Colombo,Herty,Sachers2008]. 4 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Transversality condition (TC) Let Ψ fulfill the transversality condition det [ D 1 Ψ(ˆ u ) r 2 (ˆ u 1 ) , . . . , D n Ψ(ˆ u ) r 2 (ˆ u n )] � = 0 , where ∂ D j Ψ(ˆ ∂ u j Ψ(ˆ u ) = u ) , 5 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Transversality condition (TC) Let Ψ fulfill the transversality condition det [ D 1 Ψ(ˆ u ) r 2 (ˆ u 1 ) , . . . , D n Ψ(ˆ u ) r 2 (ˆ u n )] � = 0 , where ∂ D j Ψ(ˆ ∂ u j Ψ(ˆ u ) = u ) , u ∈ R 2 n is a steady state solution to PDE-Setting (and Ψ(ˆ ˆ u ) = 0), 5 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Transversality condition (TC) Let Ψ fulfill the transversality condition det [ D 1 Ψ(ˆ u ) r 2 (ˆ u 1 ) , . . . , D n Ψ(ˆ u ) r 2 (ˆ u n )] � = 0 , where ∂ D j Ψ(ˆ ∂ u j Ψ(ˆ u ) = u ) , u ∈ R 2 n is a steady state solution to PDE-Setting (and Ψ(ˆ ˆ u ) = 0), u j ) has strictly negative λ 1 (ˆ Df (ˆ u j ) and strictly positive eigenvalue λ 2 (ˆ u j ) with linearly independent (right) eigenvectors r 1 and r 2 5 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Transversality condition (TC) Let Ψ fulfill the transversality condition det [ D 1 Ψ(ˆ u ) r 2 (ˆ u 1 ) , . . . , D n Ψ(ˆ u ) r 2 (ˆ u n )] � = 0 , where ∂ D j Ψ(ˆ ∂ u j Ψ(ˆ u ) = u ) , u ∈ R 2 n is a steady state solution to PDE-Setting (and Ψ(ˆ ˆ u ) = 0), u j ) has strictly negative λ 1 (ˆ Df (ˆ u j ) and strictly positive eigenvalue λ 2 (ˆ u j ) with linearly independent (right) eigenvectors r 1 and r 2 Corresponding characteristic fields to be either genuine nonlinear or linearly degenerate. 5 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Relevance of Transversality and Lax curves to CC By upper assumptions to the eigenvalues of the system on each arc: Ψ is implicitly R n �→ R n (Of each arc exactly one Lax curve ’enters’ the junction!). 6 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Relevance of Transversality and Lax curves to CC By upper assumptions to the eigenvalues of the system on each arc: Ψ is implicitly R n �→ R n (Of each arc exactly one Lax curve ’enters’ the junction!). This and TC ⇒ Unique solution to CC. 6 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Relevance of Transversality and Lax curves to CC By upper assumptions to the eigenvalues of the system on each arc: Ψ is implicitly R n �→ R n (Of each arc exactly one Lax curve ’enters’ the junction!). This and TC ⇒ Unique solution to CC. As an example: Subsonic states of isothermal gas dynamics fulfill those assumptions. 6 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Discretization and CFL Equidistant spatial grid x i + 1 − x i = ∆ x . 7 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Discretization and CFL Equidistant spatial grid x i + 1 − x i = ∆ x . Choose t m + 1 − t m = ∆ t such that CFL condition λ max ∆ t ≤ ∆ x holds (as usual). 7 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Discretization and CFL Equidistant spatial grid x i + 1 − x i = ∆ x . Choose t m + 1 − t m = ∆ t such that CFL condition λ max ∆ t ≤ ∆ x holds (as usual). Center of the first cell of each arc is located at x 0 = ∆ x 2 ( ⇒ boundary of each arc is located at 0 = x 0 − ∆ x 2 ) and let t 0 = 0. 7 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Finite Volume Finite Volume Method : Discretization for each u i (seperately); cell average U m i , j of u j in cell i at time t m . 8 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Finite Volume Finite Volume Method : Discretization for each u i (seperately); cell average U m i , j of u j in cell i at time t m . Evolution over ∆ t : j , i − 1 � � U m + 1 = U m ( F j ) i + 1 2 − ( F j ) i − 1 . j , i ∆ x 2 8 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Finite Volume Finite Volume Method : Discretization for each u i (seperately); cell average U m i , j of u j in cell i at time t m . Evolution over ∆ t : j , i − 1 � � U m + 1 = U m ( F j ) i + 1 2 − ( F j ) i − 1 . j , i ∆ x 2 ( F j ) i − 1 2 is flux at (left) cell boundary to cell i . 8 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Finite Volume + Coupling By the coupling condition we get boundary conditions for the PDEs at x = 0! ( ⇒ ( F j ) 0 − 1 2 ) 9 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Finite Volume + Coupling By the coupling condition we get boundary conditions for the PDEs at x = 0! ( ⇒ ( F j ) 0 − 1 2 ) The cell average at the first cell i = 0 at time t m is given by the states U m j , 0 , j = 1 , ..., n . We assume them to be sufficiently close to ˆ u j such that the TC holds. 9 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Let s → L κ ( u o , s ) the κ -th Lax curve through the state u o for κ = 1 , 2. 10 / 27
Model Numerical Scheme Coupling Condition in Scheme Algorithm Linear Case and Numerical Results Left to do and Outlook Let s → L κ ( u o , s ) the κ -th Lax curve through the state u o for κ = 1 , 2. For ( s ∗ 1 , . . . , s ∗ n ) we solve (e.g. using Newton ′ s method ) L 2 ( U m 1 , 0 , s 1 ) , . . . , L 2 ( U m � � Ψ n , 0 , s n ) = 0 first-order accurate, which is unique thanks to TC. 10 / 27
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