Numerical Discretization of Coupling Conditions by High-Order Schemes

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1 S E P T E M B E R 0 P R E P R I N T Numerical Discretization of Coupling Conditions by High-Order Schemes Mapundi K. Banda, Axel-Stefan Häck and Michael Herty Institut für Geometrie und Praktische Mathematik Templergraben, Aachen, Germany Date: September, 0. University of Pretoria, Pretoria, South Africa mapundi.banda@up.ac.za. RWTH Aachen University, Aachen, Germany; haeck@igpm.rwth-aachen.de. RWTH Aachen University, Aachen, Germany; herty@igpm.rwth-aachen.de.

2 Manuscript Click here to download Manuscript: paper_coupling.pdf 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY Abstract. We consider numerical schemes for hyperbolic conservation laws on networks coupled by possibly nonlinear coupling conditions at nodes of a network. We develop high-order finite volume discretizations including coupling conditions. The reconstruction of the fluxes at the node is obtained using derivatives of the parameterized algebraic conditions imposed at the nodal points in the network. Numerical results illustrate the expected theoretical behavior. R0, Q, F0 Keywords. numerical methods, higher-order coupling, networks of fluid dynamics,. Introduction We are interested in the high order numerical discretization of flow problems on networks where the dynamics are governed by systems of nonlinear hyperbolic partial differential equations. Among the many examples where such systems arise are traffic flow [,,, ], production networks [,, ], telecommunication networks [0], gas flow in pipe networks [,,, ] or water flow in canals [,,, ]. Mathematically, flow problems on networks are boundary value problems where the boundary value is implicitly defined by a coupling condition. This condition is either physically or mathematically motivated and may consist of algebraic conditions coupling the flow in different arcs or may consist of ordinary differential equations. We are interested in finite volume methods to resolve the hyperbolic dynamics. Most of the available finite volume methods solve the coupling condition explicitly [, ]. For the evolving state at the node, a Godunov type scheme [] or kinetic scheme [] is applied to determine the fluxes at the cell interface. Those schemes are explicit in time and therefore the dynamics on different arcs decouple contrary to an implicit discretization in time []. We develop a second order finite volume scheme for general hyperbolic systems on networks. The crucial point is the derivation of a suitable numerical flux at the nodal point where we apply high order reconstruction using temporal derivatives of the state at the node. We use a characteristic decomposition of the temporal derivative of the solution at the nodal point and estimate the outgoing information using spatial derivatives. This procedure has been applied to pure boundary value problems in the context of finite volumes for example in [ ] and mimics the Cauchy Kowalewski theorem for sufficiently smooth solutions. We show that the derived scheme coincides Date: September, 0. University of Pretoria, Pretoria, South Africa mapundi.banda@up.ac.za. RWTH Aachen University, Aachen, Germany; haeck@igpm.rwth-aachen.de. RWTH Aachen University, Aachen, Germany; herty@igpm.rwth-aachen.de.

3 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY with a second order discretization in the case of two connected arcs, we validate the discretization by reformulating a classical boundary value problem using coupling conditions and we present numerical results for gas flow in pipe networks. The only other approach to high order coupling conditions on networks which we are aware of is []. Therein, additional ghost cells for each connected arc are introduced to recover a solution and the information on the temporal derivatives is obtained without characteristic decomposition. We discuss the relation between the schemes in Section.. Motivation and numerical scheme As in [,,, ] we consider the following model for flows on graphs. This situation covers examples in gas pipe networks, traffic flow and water flow in open canals. We consider a single vertex connected to n arcs j =,...,n. The arcs extend to infinity hence they are parameterized by the interval [0,. The vertex is located at x =0for all arcs. For simplicity of the presentation, we assume that the flux f is the same on all connected arcs. Let f C R ; R andu j x, t :R + 0 R+ 0 R for j =,...,n where u j denotes the conserved variables on arc j. The dynamics of u j are governed by t u j + x fu j =0,t 0, x 0, u j x, 0 = u j,0 x, x 0, Ψu 0+,t,...,u n 0+,t = 0, t 0 where Ψ : R n R n is the possibly nonlinear coupling condition. We assume that Ψ fulfills a transversality condition below: det [D Ψûr û,...,d n Ψûr û n ] 0. which ensures well posedness of problem to. Here, D j Ψû = u j Ψû andû R n is a steady state solution to such that Ψû = 0. We refer to [, Definition.] and [, Theorem.] for more details on the definition of the solution and assumption on the characteristic fields as well as the existence and uniqueness result. Furthermore, the Jacobian of f, Dfû j, admits a strictly negative λ û j andstrictly positive eigenvalue λ û j with linearly independent right eigenvectors r and r, respectively. The associated characteristic fields are assumed to be either genuine nonlinear or linearly degenerate. Now we present the finite volume method [, ] which is employed to numerically solve -. This is essentially based on the proof of the well posedness. Consider a regular uniform grid of cell size Δx = x i+ x i,wherex i, x i+ are the cell centers and time step Δt = t m+ t m, chosen so that the CFL condition [] λ max Δt Δx, whereλ max is the largest wave speed, is satisfied. We assume the gridpoints are labeled by i, m N 0 such that x 0 = Δx and t 0 = 0. Hence, the center of the first cell i = 0 in each arc is located at x 0 and the physical node is located at the boundary x =0. For some compact domain Ω R such that u j,0 Ωandu j x, t Ω, we compute the spectral radius ρ of Df as λ max =max v Ω ρdfv.

4 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES In addition the cell boundaries interfaces are denoted x i /, on the left, and x i+/ on the right such that Δx = x i+/ x i /. Sometimes the notation I i := [x i /,x i+/ ] will be used. The discretization is undertaken component-wise for each u j. Hence the cell average of u j in cell i at time t m is denoted by Uj,i m and defined as Uj,i m := xi+ u j x, t m dx. Δx x i The evolution of the cell average over a single time-step, Δt, is Uj,i m+ = Uj,i m F j m F Δx i+ j m i where the numerical flux across the cell interface in arc j is given by t m+ F j m = fu i+ j x i+,sds. t m Remark.. The equivalent formulation of equation is: Uj,i m+ = Uj,i m Δt F j m F Δx i+ j m i where t m+ F j i+ = fu j x Δt i+,sds. t m which is referred to as the temporal integral average of fux, t at x i+/ []. In Godunov s method [] F j m = F i+ j Uj,i m,um j,i+ in which the exact solution to a Riemann problem posed at the cell boundary i + is used to define the numerical flux F j m. Thus in the original Godunov s method a piecewise constant reconstruction i+ u j x, t m = Uj,i m χ [xi,x i+ ]x, i where the characteristic function χx is defined in the usual way as {, if x [xi,x χ [xi,x i+ ]x = i+ ] 0, otherwise, was applied as a numerical approximation for u j x, t attimet m. To generalize Godunov s method, the piecewise constant approximation of the solution can be replaced by a more accurate representation. In this paper, we consider a piecewise linear approximation. Thus the reconstruction in takes the form u j x, t m = P i x, t m ; Uj, m χ [xi,x i+ ]x, i where P i x, t m ; Uj, m is a linear reconstruction in cell I i using data {Uj, } m in arc j such that P i x, t m ; Uj, m =Uj,i m + σj,ix m x i for x I i. The slope σj,i m in cell I i is also based on the data {U j, }. m Note that σm j,i =0 recovers the first-order Godunov method.

5 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY To reduce oscillations, a slope limiter is applied to σj,i m []. A simple choice of slopes is the so called minmod slope: σj,i m = Δx minmodu j,i+ m Uj,i,U m j,i m Uj,i m where the minmod function is defined as: a, if a < b and ab > 0; minmoda, b = b, if b < a and ab > 0; 0, if ab 0; = sgna+sgnb min a, b. It can be noted that, just as in first-order Godunov method, a Riemann problem needs to be solved at the cell boundaries x i+/ since the reconstruction provides two values at the cell interfaces x i+/ whichwedenoteby u j x,t m =U i+ j,i ; m u j x +,t m =U i+ j,i+ m which are values based on the polynomial P i x x=xi+ on the left of the interface and P i+ x x=xi+ on the right of the interface, respectively. The coupling condition at the vertex induces a boundary condition for equation at x =0. Attimet m the cell averages in the first cell i = 0 of the connected arcs j are given by Uj,0 m for all arcs j =,...,n. Assume that they are sufficiently close to û j such that condition holds. Denote the κ th Lax curve through the state u 0 for κ =,. by s L κ u o,s. Then solve for s,...,s n using, for example, Newton s method the nonlinear system Ψ L U,0,s m,...,l Un,0,s m n =0. A unique solution exists due to. The boundary value Uj,0 m+ at time t m+ is then given by equation for i = 0 and with Uj, m := L Uj,0,s m j, j {,...,n}. This construction yields a first order approximation to the coupling condition and the solution u j. In order to obtain at least a first order convergent scheme, we consider a reconstruct, evolve and average algorithm []. The finite volume formulation of equation is given by equation and equation. We proceed as follows: STEP Given the cell averages Uj,i m, we reconstruct a piecewise linear function u jx, t m on each cell [x i,x i+ ]. This is a standard procedure and more details can be found, for example, in [0, ]. We apply a reconstruction with slopes obtained by the minmod limiter for the cells i =,...,as discussed for equation above. In the first cell i = 0, we modify the reconstruction of the slope due to the absence of cell averages beyond the vertex since σj,0 m = Δx minmodu j, m Uj,0,U m j,0 m Uj,. m

6 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES Instead, for Uj, m the procedure applied for the first-order case can be applied here, see equation. Denote the vector of the recovered slopes for both components by σ j,i = σj,i,σ j,i fori>0. Using the above construction, we obtain the reconstruction u j x, t m =σ j,i x x i +U m j,i, x i x x i+,i=,..., The previous reconstruction provides two values at the cell interfaces x i+ from cell i and denoted in the following by x i+ i +, respectively. for the reconstruction at x I+ STEP Reconstruct for each arc j a piecewise linear function v j t fort m t t m+ such that Ψv t m,...,v n t m = 0. and d dt Ψv t,...,v n t t=t m =0. STEP Evaluate equation using a predictor corrector step [] for all cells except for i =0. This approach is also attributed to Hancock in [, ]. We split the flux as in a Lax Friedrichs scheme fu =f + u+f u := fu+au+ fu au where a = λ max. Due to splitting the wave speeds of the fluxes fu ± au do not change sign across x i+. Using this fact and the midpoint rule at time t m+ = t m + Δt, the evolution of the flux is given by F j i+ =Δt f + u j x i+,t m+ +f u j x i+ +,t m+ + OΔt. Using Taylor expansion and the linear reconstruction, we obtain up to second order in space and time Uj,i m := Uj,i m Δx + σ j,i Δx F j i+ Δt Δx,Um j,i+ := Uj,i+ m Δx σ j,i+, f + Uj,i m Δt f U m j,i+ Δt DfU m j,i+σ j,i+ DfU j,i σ m j,i. STEP Evaluate the fluxes in cell i =0. The only flux which needs to be evaluated is F j. Due to the construction of the boundary conditions in STEP the characteristic speed of information is non negative. Therefore, F j = t m+ fu j x,sds t m t m+ + fv j sds =Δtfv j t m+ + OΔt. t m

7 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY Similarly, to STEP we evaluate Δx F j Δt Δx f v j t m + Δt d dt v jt t=t m STEP Evolve the dynamics according to equation for i =,..., to obtain the new cell averages at time t m+ and proceed with STEP. To approximate the slopes σ j,0 we first calculate the p.w. constant information U j, as given by. This data can now be used to gain the σ j,0 in the usual manner see STEP. This approach preserves the Total variation diminishing-property of the scheme. The difference to a standard first order method is STEP and STEP. In order to determine the values v j t m, we proceed similarly to the discussion in equation. However, since we reconstruct the function u j x, t m onarcj by a piecewise linear function, the value of u j x, t m closest to the vertex at time t m is given by u j x,t m =Uj,0 m Δx σ j,0. Hence, we determine the vector s =s,...,s n R n by solving the possibly nonlinear equation. Ψ L U,0 m Δx σ,0,s,...,l Un,0 m Δx σ n,0,s n =0. For Uj,0 m sufficiently close to û and due to condition, the previous equation has a unique solution s. Hence we define v j t m :=L Uj,0 m Δx σ j,0,s j. For the reconstruction in equation, we additionally need at least to recover also the slope d dt v jt attimet = t m. Condition is supposed to hold true also for t t m. Hence, we obtain for sufficiently smooth solutions 0= d dt Ψu +,t,...,u n 0+,t =. n D uk Ψu 0+,t,...,u n 0+,t t u k 0+,t. k= In a neighborhood of û the eigenvalues have a sign and, therefore, only a part of the information of t u k 0+,t is available at the vertex. Further, due to we will have waves emerging from the vertex in short time. Therefore, in of D uk Ψu 0+,t,...,u n 0+,t we have u k 0+,t=v k t fort>t m. We now approximate t u k 0+,tfort>t m using a Roe type scheme and compute the derivative A j of f at the new position v j t m. Let the constant matrix A j be defined by A j := Dfv j t m, j=,...,n. Since f is strictly hyperbolic and for v j t m in the neighborhood of û, each A j is diagonalizable with eigenvalues λ j < 0andλ j > 0 and corresponding linearly independent set of eigenvectors r j, = r v j t m and r j, = r v j t m, respectively. Let R j := r j,,r j,.

8 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES For small values of t t m 0, we approximate u j 0+,t by a decomposition in eigenvectors v u j t, 0+ vj t r j, + vj t r j, =: R j t j vj t, Further, we approximate the dynamics of t v j, t for small t t m by t vj t m = λ j x Rj u j t m, 0+ = λ σ j Rj j,0. where we used the linear reconstruction of u j x, t onarcj and the super index denotes the component {, } of the vector. Then, we obtain from equation at time t m an equation in the unknowns vj n v 0= D k Ψv t m,...,v n t m R j t j vj t k= or equivalently n D k Ψv t m,...,v n t m r v k t m λ k 0 k= R k σ j,0 σ k,0 = σ k,0 n D k Ψv t m,...,v n t m r v k t m t vk tm k= Again, due to condition the previous equation has a unique solution for the n values t vj tm,j=,...,n. Then, we set d dt v jt m := t vjt l m r l v j t m. l= A piecewise linear reconstruction of v j yields at time t = t m Ψ v t m + d dt v t m t t m,...,v n t m + d dt v nt m t t m = Ot t m. We summarize the computations in the following Lemma. Lemma.. Consider a single node with n connected arcs and let t m be some positive time. Let Ψ C R n ; R n and let û j := Uj,0 m Δx be such that equation holds true. Then, for v j t as in the previous construction, the coupling condition is satisfied up to second order in time, i.e. Ψv t,...,v n t = Ot t m.. Properties of the scheme in the linear case In order to illustrate the properties of the scheme, we first consider the case of a single arc extending from to. We drop for a moment in the index j. Also, assume that u R, f: R R and is given by fu =cu, c>0.

9 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY Since a = c, we obtain f + u =cu, f u =0andDf ± u = c ± a. condition gives cδt Δx. The numerical flux at the cell boundary is then given by Ui m. F i+ = Δt Δx c The scheme, therefore, takes the form: Ui m+ = Ui m Δt Δx c Ui m + σ i Δx cδt σ i U m i + Δx cδt σ i σ i. The CFL One easily verifies that this method is second order accurate in space and time. Further, we consider the scalar case n = and two connected arcs j =, withf u = c u, f u =cuand c>0. Notethatthiscasedoesnot fulfill the general assumption on the eigenvalues given previously. Therefore, we need to modify the construction slightly. However, we still want to treat this case to show its relation to a discretization of a single scalar advection equation. The coupling condition Ψ preserves the total mass and reads Ψu 0+,t,u 0+,t = u 0+,t u 0+,t=0. This situation is then equivalent to a Cauchy problem for yx, t with combined domain x R and t 0 u,0 x, for x 0 t yx, t+c x yx, t =0,yx, 0 = y 0 x :=. u,0 x, for x>0. In the following derivations, we show that the previous construction leads to a scheme of second order in space and time for equation. Clearly, the eigenvalues are λ = c and λ = c for arcs j =,, respectively, and therefore for arc j =,the assumptions are not fulfilled. Therefore, at x = 0, we do not prescribe any boundary condition. Hence, a single coupling condition is sufficient to have a well posed problem and initial data. t u j + x f j u j =0,x 0, t 0, and u 0+,t u 0+,t=0,t 0. In order to use a similar notation as above, we note that the admissible boundary states for arc j =aregivenby L û,s =s +û,s R. In arc j =,wehavel û,s =û. Hence, for the unknown s the condition becomes: U,0 m Δx σ,0 U,0 m Δx σ,0 + s =0. and equation gives: v t m =U,0 m Δx σ,0, v t m =U,0 m Δx σ,0. We can not use condition directly, since the assumptions on the sign of the eigenvalues are not fulfilled in arc j =. However, we can repeat the computations to and obtain, for sufficiently smooth solution u 0+,t, 0= t u 0+,t t u 0+,t=+c x u 0+,t R = cσ,0 R.

10 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES Here,wealsousedthefactthatonarcj = and for sufficiently smooth u,wehave t u 0+,t c x u 0+,t=0. Therefore, the linear reconstruction of v t fort m+ > t t m reads v t =U,0 m Δx σ,0 + cσ,0 t t m. The numerical flux in cell i = 0 is then given as F =Δtc U,0 m Δx σ,0 + Δt cσ,0. In order to compare the proposed method with a numerical discretization of equation it suffices to consider the discretization of the first cell i = 0 of arc j =. The remaining cells are independent of the coupling condition and therefore the applied discretization coincide. In the first cell i = 0 of arc j = we obtain from and 0 U,0 m U,0 m +σ,0 + σ,0. U m+,0 = U m,0 Δt Δx c Δx cδt Suppose a discretization of equation with i N and such that at x =0wehavethe location of the physical node and as before x i = Δx for i = 0 is the location of the center of the grid. The discretization 0 has to be compared with a corresponding second order discretization of equation in cell i =0, i.e., for x [ Δx, Δx ]. Using equation we obtain for the cell averages Yi m = xi+ yx, t m dx Δx x i Y0 m+ = Y0 m Δt Δx c Y0 m Y m Δx cδt +τ 0 τ. The construction of σ,0 and τ 0 are independent of the coupling condition and therefore σ,0 = τ 0. In the continuous case we have y x, t = u x, t and therefore Yi m = U, i+ m for i<0and the slopes of the linear reconstruction of y and u are related as σ, i = τ i+, i < 0. Hence, provided we use the same derivation of the slopes for U j,i and Y i we observe that the proposed discretization of the coupling condition leads to the same scheme as a discretization for the Cauchy problem. In particular, the coupling condition does not lead to an order reduction. We summarize the findings in the following Lemma. Lemma.. Let n =and consider for x, t R + 0 R+ the problem t u + c x u =0, t u c x u =0, with initial data given by and coupling condition. On an equidistant spatial grid x i,i N consider a piecewise linear reconstruction u j x, t m for j =,. Consider furthermore the second order MUSCL discretization of equation with initial data at time t = t m given by yx, t m =u j x, t m H x+u j x, t m Hx. Then, for c = a and j 0, the cell averages U,j m+ and U,j m+ obtained from the proposed algorithm coincide with the cell averages Y m+ j and Y m+ j, respectively. Next, we consider the case of a linear system fu =Au with A R strictly hyperbolic with λ < 0andλ > 0. We denote by R =r,r the matrix of the right

11 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY eigenvectors to A. Further, consider the case of a single arc n =. Then, problem is a boundary value problem for u x, t withx 0 and boundary conditions induced by equation. We prescribe as boundary condition t bt,b C, in the second characteristic variable. In terms of Ψ, we obtain Ψu 0+,t = R u 0+,t bt. On the time interval t m,t m+ we expect the linear construction in the second component v t R to be vt =R R u 0+,t m + t u 0+,t m,bt m +b t m t t m T + Ot t m. We compare the presented approach to the approach presented in []. Due to the linearity of A, we have for any û R Lû, s =û + sr. At first we discuss the reconstruction of STEP. Due to the linear reconstruction in each cell, we have U,0 m Δx σ,0 = u 0+,t m +OΔx. Equation, and yield R U,0 m Δx σ,0 + s r bt m 0=ΨL U,0 m Δx σ,0,s = = R U,0 m Δx σ,0 + s bt v t m =U,0 m Δx σ,0 + s r = R R U,0 m Δx σ,0 = R R U,0 m Δx σ,0 R = R u 0+,t m + OΔx bt m. + bt m r R U,0 m Δx R U m,0 Δx σ,0 0 0 σ,0 R + R 0 bt m The slope d dt v t m is computed using equation to. Equation reads 0 = R t u 0+,t b t, and we approximate t u 0+,t by decomposition in characteristic variables to obtain t v t m = λ R σ,0 = R t u 0+,t m + OΔx. and according to at time t m 0= R R t u 0+,t m r + t v t m r b t m = t v t m b t m.

12 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES Due to, we obtain for t t m = OΔx the reconstruction R v t =R u 0+,t m bt m +t t m b t m r + R t u 0+,t m r R u = R 0+,t m +t t m t u 0+,t m + OΔx bt m +t t m b t m. In [] the following approach has been proposed to obtain a high order reconstruction of v t. We apply this procedure to a first order scheme studied herein. The derivation of v t m is as above. However, there is a difference in the approximation of the derivative t v t m. Instead of a characteristic decomposition, the value of the derivative is approximated using the Lax curve to the linearized system, i.e., L t u 0+,t,s t = t u 0+,t+s tr, and the real valued unknown s t attimet = tm is obtained as solution to equation. In order to evaluate t u 0+,t the linearized equation, i.e., t u 0+,t m = λ 0 0 λ σ,0 is used leading to an error of order OΔx due to the spatial reconstruction of u x, t. Hence, 0= R t u 0+,t b t m +s t m R r + OΔx Finally, the reconstruction of v t up to second order in time is given by R v t =R u 0+,t m bt m +t t m L t u 0+,t,s t R = R u 0+,t m +t t m t u 0+,t m + OΔx bt m +b t m t t m +OΔx In characteristic variables Rv t, the difference of and is of order OΔx in the recovered boundary condition. This is an error of the order of the scheme. However, in the presented approach no information on R t u 0+,t being the outgoing characteristic is needed and for small Δt the resolution of bt isoforder Δt instead of order Δx = λ maxδt. The previous relation also holds true in a more general setting: Consider n connected arcs and denote the state at x =0+onarck by û k = Uk,0 m Δx σ k,0. Denote by A k = Dfû k andbyr k =r k,,r k, the matrix of right eigenvectors of A k. The superscript in u l denotes the component l of the respective vector u. Let û k,t be û k,t := Aσ k,0 = t û k 0+,t m +OΔx and let Ψ : R n R n be a C function with partial derivatives Ψ uk := D uk Ψû. Then, [] requires to solve the following system for ξ =ξ k k R n and reconstruct vt R n by n 0= Ψ uk û k,t + ξ k r k,, k= d dt v kt m :=û k,t + ξ k r k,.

13 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY Due to condition the matrix A R n n defined by A =Ψ u r,,...,ψ un r n, is invertible and we obtain for the second component of d dt v jt m and n j ξ j = A Ψ uk û k,t the following equivalence k= Rj d dt v jt m = Rj û j,t ξj Rj r j, n ûk,t j = Rj û j,t A Ψ uk R k R k = k= = Rj û j,t n j n A Ψ uk r k, R k ûk,t + Ψ uk r k, R k ûk,t k= k= j n = Rj û j,t A Ψ uk r k, R k ûk,t k= R û,t A A. Rn û n,t Summarizing, we observe that the previous computation of the second component of d dt v jt m uptooδx with equation and. Hence, the proposed method slightly improves the construction presented in []. Compared with [] it also does not require during the construction information on Rj û j,t being the information on the outgoing characteristic on arc j.. Application to gas dynamics We discuss the application of the method to gas dynamics in connected pipe systems. Those problems have been studied intensively in the past years and we refer e.g. to [,,, ] for analytical and computational results. Here, we study a single node with n connected arcs and on each arc the dynamics are governed by the isentropic Euler equations or the p-system. We assume a subsonic state û =û j n j= is given such that condition is fulfilled. The wave curves and properties of the isentropic Euler equations are well known [] and they are given on arc j by 0= t ρ j + x q j and 0 = t q j + x pρ j + q j. ρ j Here, p C is the pressure law which is supposed to be non negative, strictly monotone increasing and convex, with e.g. pρ =a ρ being a model for isothermal j

14 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES gas. The subsonic region is the set Ω := {ρ, q :ρ>0,λ ρ, q < 0 <λ ρ, q} where λ, = q ρ p ρ. The corresponding right eigenvectors are r ρ, q = λ ρ, q and r ρ, q = The reversed Lax curve exiting at ˆρ, ˆq isgivenby L û, σ = σ ˆqˆρ + σˆρ σ ˆρpσ pˆρ, σ σ ˆqˆρ σ ˆρ σ λ. ρ, q > ˆρ. p s s ds, σ ˆρ Different coupling conditions have been proposed in the literature []. Common to all is the conservation of mass flux across the node. Hence, the first component of Ψ reads Ψ ût, 0+ = n ˆq j t, 0+. The remaining n conditions can be prescribed for example by the equality of the pressure j= Ψ j û0+,t = pˆρ j 0+,t pˆρ j 0+,t, j=,...,n. Remark.. Let n =. Consider coupling condition and the equality of momentum Ψ û0+,t = pˆρ 0+,t + ˆq 0+,t ˆρ 0+,t pˆρ +,t ˆq 0+,t ˆρ 0+,t. We recover a solution ρ, qx, t to the Cauchy problem 0= t ρ + x q, 0= t q + x p ρ+ q ρ,x R, t 0, by qx, t = q x, t and by qx, t =q x, t for x R + 0. For details we refer to []. For the reconstruction of v j t in STEP, we collect the following elementary computations. Equation is solved for s R n using Newton s method with L û, s given by equation and Ψ given by and, respectively. This enables us to determine v j t m according to. In order to obtain t v j t m, we solve equation with R j =r v j t m,r v j t m. For the second component we note that D u Ψûr û,...,d un Ψûr û n λ ˆρ, ˆq λ ˆρ, ˆq... λ ˆρ n, ˆq n p ˆρ p ˆρ = p ˆρ p ˆρ.... p ˆρ n p ˆρ n

15 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY Further, for û := v j t m n j= we compute the vector n D k Ψûr û k λ û k R k σ k,0 k= λ ˆρ, ˆq λ ˆρ, ˆq... λ ˆρ n, ˆq n p ˆρ p ˆρ = p ˆρ p ˆρ... p ˆρ n p ˆρ n λ û R σ,0. λ û n R n σ n,0. The previous computations enable us to solve equation for t v k tm and hence to determine d dt v jt m by equation.. Computational results.. Linear and node vs linear on a single arc. Here we will show numerical results on a linear transport equation as for a single arc therefore, we drop the index j. The setup is a periodic domain x [0, π] i.e. for the state u we have u0,t = uπ, t, t 0, as flux we set f, and the initial condition is picked as u o x =sinx. For the nodal solution on the domain [0, π] we coupled the arc with itself and we used φ as in, i.e. Ψut, 0+,ut, π = ut, 0+ ut, π =0 t 0. Furthermore, the Courant number is set to 0. and the following examples are grounded on a final time T =. For both scenarios we implementet a first order accurate Upwind scheme as well as a MUSCL scheme [], where we used the modified MUSCL scheme as in 0 in the coupled case. The following will illustrate both scheme in the spacial cyclic and in the nodal coupled case. The power k of N k = k the number of degrees of freedom of the spacial discretization we plotted against loge N k log,wheree N k is the L error to the analytical solution for agivenn k. bla

16 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES Figure. L error of both, periodic boundary and nodal coupled method numerical approximation to analytical solution. Here, table. contains the L errors and convergence rate of the upper plots. N k and r k = loge N k log as above. The Rate columns are therefore r k r k+ of each error. N k Upwind r k Rate MUSCL periodic r k Rate MUSCL nodal r k Rate 0. /.0 /. / Table. Comparison of convergence of the periodic boundary to the nodal coupled method.. Gas dynamics. We compute the conservation law given by equation. As above, we have a spatial domain x [0, π] withu0,t=uπ, t t 0 and we are going to couple this arc with itself. We compare the results with a second order MUSCL scheme on a periodic domain. As first component of the coupling Ψ the conservation of mass is used. For the second component we employ the conservation of momentum

17 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY as suggested in Remark.. The second condition leads to a true nonlinear coupling compared with the approach by []. There, a linear coupling is used. Further note that we are connecting the states of the arc at spatial coordinates x =0 and x =π. Therefore we have to alter the notion on the Lax-curves. At x =0weuse the second Lax-curve at stated in. In contrast to this at x = we have to employ the first Lax-curve. A parametrization of this can be found in []. The initial data we set are ρx, 0 0. cosx+ Ux, 0 = =. qx, cosx+ The final time is T =0. to prevent shocks from happening to recover the second order accuracy of the scheme. The chosen pressure law is pρ =a ρ and a =. Again a MUSCL scheme is used as in Section suggested. We can compare the periodic standard second order solution U cyc = ρ cyc,q cyc to the coupled solution U cc =ρ cc,q cc. Figure. Periodic versus nodal coupled solution The Table. contains the L errors and convergence rate of the coupled solution compared to the periodic case. N k = k is the number of degrees of freedom due to the spatial discretization. The column L ρ givesr ρk = log ρ cyc ρ cc L / log and L q r qk = log q cyc q cc L / log analogously. The Rate columns are therefore r k r k+ of each error of both quantities ρ and q.

18 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES N k L ρ L q Rate ρ Rate q.. / / Table. L convergence of the periodic boundary to the nodal coupled method The Table. contains the L errors and convergence rate of the coupled solution. The scenario is the same as in Table. but all data are presented under the L norm. We observe here, that the scheme is first order only. This is due to the local first order accuracy of the TVD method at a local extremum as reasoned in chapter of []. N k L ρ L ρ Rate ρ Rate q.. / / Table. L convergence of the periodic boundary to the nodal coupled method.. Gas dynamics Y-junction. Here we have a look at the setup as suggested in Section for n = number of arcs attatched to the node at x = 0. Obviously, in the numerics we cannot extend the arcs to infinity. Therefore we consider a finite spatial domain, which will be x [0, ] for all arcs. For the boundary which is not adjacent to the coupled knot we implemented Neumann boundary conditions. The chosen coupling conditions are the conservation of mass and conservation of momentum.. The initial data are as follows: For the first arc j = we set ρ x, 0 U x, 0 = q x, 0 with ρ x, 0 = { x + x + x [0, x [, ]

19 0 0 MAPUNDI K. BANDA, AXEL-STEFAN HÄCK, AND MICHAEL HERTY and { x + q x, 0 = x x [0, x [, ]. For j =, wehave ρj x, 0 U j x, 0 = q j x, 0 with q j x, 0 0 and ρ j x, 0. Those initial data are smooth and give us a feasible initial state with respect to the coupling condition. Hence, ΨU x, 0,U x, 0,U x, 0 = 0, 0, 0. The chosen pressure law is again pρ =a ρ and a = and the final time is set to T =0.. The remaining numerics is the same as in.. Figure. Evolution of Density ρ. Arcs: j =, j =, j =. The Table. contains the L errors and convergence rate of the coupled solution compared to a numerical solution of higher discretization at each arc j =,,. The errors and order are computed analogously to table.. Since the initial data in arc j = and j = are identical and the coupling handles all arcs the same way we get coniciding data for those both arcs. hence, we compress both data sets to one in the convergence tabel.

20 0 0 NUMERICAL DISCRETIZATION OF COUPLING CONDITIONS BY HIGH-ORDER SCHEMES Figure. Evolution of Flow q. Arcs: j =, j =, j =. j =: N k L ρ L q Rate ρ Rate q..0 / / j =, : N k L ρ L q Rate ρ Rate q.0. / / Table. L convergence on a network with arcs Acknowledgements. This work has been partly supported by DFG excellence initiative and the excellence cluster EXC, BMBF KinOpt and DAAD. This work is also based on the research supported in part by the National Research Foundation of

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