well-posedness for some lwr models on a junctionacesar/slidesanconet/andreia...introductionmonotone...

Introduction Monotone junction solvers BLN conditions via Godunov flux Admissibility conditions. Germ GVV . Well-posedness

Well-posednessfor some LWR models on a junction

Boris Andreianov1,in collaboration with

Guiseppe M. Coclite2 and Carlotta Donadello3

1Laboratoire de Mathématiques et Physique ThéoriqueUniversité de Tours, France

2Dipartimento di Matematica, Università di Bari, Italia3Laboratoire de Mathématiques de Besançon

Université de Franche-Comté, France

ANalysis and COntrol on NETworks: trends and perspectivesPadua, March 9-11, 2016

Statement of the problem

We consider a junction where m incoming and n outgoing roads meet.

Incoming roads: x ∈ Ωi = R−, i = 1, . . . ,m ;Outgoing roads: x ∈ Ωj = R+, j = m + 1, . . . ,m + n ;The junction is located at x = 0.

On each road the evolution of traffic is described by

∂tρh + ∂x fh(ρh) = 0, h = 1, . . . ,m + n,

ρh density of vehicles, [0,R]-valued for all hfh bell-shaped, non linearly non degenerate,Lipschitz fluxes

∀h fh(0) = 0 = fh(R)

Moreover, we postulate conservativity at the junction:

m∑i=1

fi (ρi (t ,0−)) =m+n∑

fj (ρj (t ,0+)).

On each road the evolution of traffic is described by

∂tρh + ∂x fh(ρh) = 0, h = 1, . . . ,m + n,

ρh density of vehicles, [0,R]-valued for all hfh bell-shaped, non linearly non degenerate,Lipschitz fluxes

∀h fh(0) = 0 = fh(R)

Moreover, we postulate conservativity at the junction:

m∑i=1

fi (ρi (t ,0−)) =m+n∑

fj (ρj (t ,0+)).

“Solutions”

Fix ~ρ0 = (ρ10, . . . , ρ

m+n0 ) s.t. ρh

0 ∈ L∞(Ωh, [0,R]), ∀h ∈ 1, . . . ,m + n.We call “solution” a (m + n)-uple ~ρ = (ρ1, . . . , ρm+n) s.t.

∀h ρh is a Kruzhkov entropy solution in R+ × Ωh \ ∂Ωh.Namely ∀k ∈ [0,R] and ∀ξ ∈ C1

c (R+ × Ωh), ξ ≥ 0∫R+

∫Ωh

|ρh − k |ξt + qh(ρh, k)ξx dx dt ≥ 0

(with qh(ρh, k) = sign(ρh − k)(fh(ρh)− fh(k)) the Kruzhkov entropy flux)

Idea: k is an obvious solution...the above inequalities are “Kato inequalities” between ρh and kconservation at the junction holds.

There is no hope to prove well-posedness for “solutions”.Analogy:junction coupling conditions (JCC) play the role of boundaryconditions (BC). Imposing mere conservativity condition as JCCleaves the Cauchy problem underdetermined ! [A. ESAIM Proc.’15].

“Solutions”

Fix ~ρ0 = (ρ10, . . . , ρ

m+n0 ) s.t. ρh

0 ∈ L∞(Ωh, [0,R]), ∀h ∈ 1, . . . ,m + n.We call “solution” a (m + n)-uple ~ρ = (ρ1, . . . , ρm+n) s.t.

∀h ρh is a Kruzhkov entropy solution in R+ × Ωh \ ∂Ωh.Namely ∀k ∈ [0,R] and ∀ξ ∈ C1

c (R+ × Ωh), ξ ≥ 0∫R+

∫Ωh

|ρh − k |ξt + qh(ρh, k)ξx dx dt ≥ 0

(with qh(ρh, k) = sign(ρh − k)(fh(ρh)− fh(k)) the Kruzhkov entropy flux)

Idea: k is an obvious solution...the above inequalities are “Kato inequalities” between ρh and kconservation at the junction holds.

There is no hope to prove well-posedness for “solutions”.Analogy:junction coupling conditions (JCC) play the role of boundaryconditions (BC). Imposing mere conservativity condition as JCCleaves the Cauchy problem underdetermined ! [A. ESAIM Proc.’15].

Many different approaches to single out “suitable” solutions

For the Riemann problem at the junction(road-wise constant initial conditions):

[Holden, Risebro SIMA’95]maximize a concave “entropy” function at the junction ;[Coclite, Piccoli SIMA’02], [Coclite, Garavello, Piccoli SIMA’05]traffic distribution matrix + optimization ;[Lebacque ’96], [Lebacque, Khoshyaran ’02]Supply-Demand model ;...

We prove well-posedness for solutions to the Cauchy problemwhich are limits of vanishing viscosity (VV) approximations.

Essential ingredient: intrinsic characterization of VV limits(a notion of solution, expressed e.g. via some “entropy inequalities”)

VV limits obey rather artificial JCC... but the study is instructive !

Many different approaches to single out “suitable” solutions

For the Riemann problem at the junction(road-wise constant initial conditions):

[Holden, Risebro SIMA’95]maximize a concave “entropy” function at the junction ;[Coclite, Piccoli SIMA’02], [Coclite, Garavello, Piccoli SIMA’05]traffic distribution matrix + optimization ;[Lebacque ’96], [Lebacque, Khoshyaran ’02]Supply-Demand model ;...

We prove well-posedness for solutions to the Cauchy problemwhich are limits of vanishing viscosity (VV) approximations.

Essential ingredient: intrinsic characterization of VV limits(a notion of solution, expressed e.g. via some “entropy inequalities”)

VV limits obey rather artificial JCC... but the study is instructive !

Vanishing viscosity approximations, Coclite-Garavello 2010

Fix ε > 0. Consider convection-ε-diffusion + JunctionCouplingCondition:ρεh,t + fh(ρεh)x = ερεh,xx ,∑m

(fi (ρεi (t ,0))−ερεi,x (t ,0)

)=∑m+n

(fj (ρεj (t ,0))−ερεj,x (t ,0)

ρεh(t ,0) = ρεh′(t ,0),

ρεh(0, x) = ρ0h,ε(x),

where the approximated initial conditions ~ρ0,ε satisfy

ρ0h,ε ∈W 2,1(Ωh) ∩ C∞(Ωh), [0,R]-valued,

ρ0h,ε −→ ρ0

h, a.e. and in Lp(Ωh), 1 ≤ p <∞, as ε→ 0,∥∥ρ0h,ε

∥∥L1(Ωh)

≤∥∥ρ0

∥∥L1(Ωh)

,∥∥(ρ0

h,ε)x∥∥

L1(Ωh)≤ TV (ρ0

ε∥∥(ρ0

h,ε)xx∥∥

L1(Ωh)≤ C0,

with C0 > 0 independent from ε, h.

Coclite and Garavello, 2010

Theory of semigroups⇒ ∀ε > 0 there exists a unique ~ρε s.t.

ρεh ∈ C([0,∞); L2(Ωh)) ∩ L1loc((0,∞); W 2,1(Ωh)), h ∈ 1, . . . ,m + n,

0 ≤ ρεh ≤ R,m+n∑h=1

‖ρεh(t , ·)‖L1(Ωh) ≤m+n∑h=1

∥∥ρ0h

∥∥L1(Ωh)

, ∀t ≥ 0,

+ additional a priori estimates like ‖√ερεx‖L2 ≤ C.

Compensated compactness⇒ existence of a sequence ε``∈N,ε` → 0 and a solution ~ρ of the inviscid Cauchy problem such that

∀h ρε`h −→ ρh, a.e. and in Lploc(R+ × Ωh), 1 ≤ p <∞. (1)

Uniqueness of VV solutions for the inviscid problem ?It is proved [Coclite, Garavello SIMA’10] in the case m = n, fh ≡ f

based on comparison with the obvious solution ~k = (k , . . . , k).

“Local objective”: extend these results to general junctions“Global objective”: better understand solvers for different JCC

Coclite and Garavello, 2010

Theory of semigroups⇒ ∀ε > 0 there exists a unique ~ρε s.t.

ρεh ∈ C([0,∞); L2(Ωh)) ∩ L1loc((0,∞); W 2,1(Ωh)), h ∈ 1, . . . ,m + n,

0 ≤ ρεh ≤ R,m+n∑h=1

‖ρεh(t , ·)‖L1(Ωh) ≤m+n∑h=1

∥∥ρ0h

∥∥L1(Ωh)

, ∀t ≥ 0,

+ additional a priori estimates like ‖√ερεx‖L2 ≤ C.

Compensated compactness⇒ existence of a sequence ε``∈N,ε` → 0 and a solution ~ρ of the inviscid Cauchy problem such that

∀h ρε`h −→ ρh, a.e. and in Lploc(R+ × Ωh), 1 ≤ p <∞. (1)

Uniqueness of VV solutions for the inviscid problem ?It is proved [Coclite, Garavello SIMA’10] in the case m = n, fh ≡ f

based on comparison with the obvious solution ~k = (k , . . . , k).

“Local objective”: extend these results to general junctions“Global objective”: better understand solvers for different JCC

Monotonicity of the VV solver

Not only the viscous problem of Coclite-Garavello is well posed.The key property, independent of ε, can be expressed as :

(i) monotonicity (order-preservation) of the solver:~ρ0 ≥ ~ρ0 (componentwise) ⇒ ∀t > 0 ~ρ ε(t , ·) ≥ ~ρ ε(t , ·);

(ii) L1-contractivity of the solver:∑m+n

h=1‖ρεh(t , ·)− ρεh(t , ·)‖L1(Ωh) ≤

∑m+n

h=1‖ρ0

h,ε − ρ0h,ε‖L1(Ωh);

(iii) Kato inequality: for all test function ξ ∈ D((0,+∞)× R), ξ ≥ 0

−∫R+

∫Ωh

(|ρεh − ρεh|ξt + qh(ρεh, ρεh)ξx + ε|ρεh − ρεh|xξx ) ≤ 0,

Links: (iii)⇒ (ii) (with ξ ∼ 11[0,t]); (ii)⇔ (i) (Crandall-Tartar lemma)

These properties are inherited by VV admissible solutions.

Kato inequality

guarantees uniqueness of VV solutionsprovides their intrinsic characterization

h=1‖ρεh(t , ·)− ρεh(t , ·)‖L1(Ωh) ≤

∑m+n

h=1‖ρ0

−∫R+

∫Ωh

Kato inequality

h=1‖ρεh(t , ·)− ρεh(t , ·)‖L1(Ωh) ≤

∑m+n

h=1‖ρ0

−∫R+

∫Ωh

Kato inequality

Monotonicity of the Riemann solver at junction

At least heuristically, JCC⇔ Riemann solver at junction.Different solution semigroups for LWR on networks originatefrom different Riemann solvers at junction (Garavello, Piccoli,. . . )

Byproduct of our analysis:

a subclass of these semigroups shares key features of VV solutions

Required property: monotonicity of the junction Riemann solver(larger data on a road lead to larger solutions on the whole network)

General principle:monotone, Lipschitz Riemann solver at junction

=⇒ an intrinsic notion of solution + well-posedness.

Notion of solution and uniqueness:mimic tools developed for discontinuous-flux scalar conservation laws[A., Karlsen, Risebro ARMA’11]: admissibility germ, adapted entropiesConstruction of solutions:approximations, e.g. by the Godunov finite volume scheme

Godunov’s numerical flux

Consider the Riemann problem for pure SCL:ut + f (u)x = 0, (t , x) ∈ R+ × R

u0(x) =

a, if x < 0,b, if x > 0.

Denote by R[a,b] its Kruzhkov entropy solution.The Godunov flux is the function (a,b) 7→ f (R[a,b])|x=0± .Analytically

G(a,b) =

mins∈[a,b] f (s) if a ≤ b,maxs∈[b,a] f (s) if a ≥ b.

Key properties of the Godunov flux:Consistency: for all a ∈ [0,R], G(a,a) = f (a);Monotonicity and Lipschitz continuity:∃L > 0 : ∀(a,b) ∈ [0,R]2 there holds

0 ≤ ∂aG(a,b) ≤ L, −L ≤ ∂bG(a,b) ≤ 0.

Godunov’s numerical flux

Consider the Riemann problem for pure SCL:ut + f (u)x = 0, (t , x) ∈ R+ × R

u0(x) =

a, if x < 0,b, if x > 0.

Denote by R[a,b] its Kruzhkov entropy solution.The Godunov flux is the function (a,b) 7→ f (R[a,b])|x=0± .Analytically

G(a,b) =

mins∈[a,b] f (s) if a ≤ b,maxs∈[b,a] f (s) if a ≥ b.

Key properties of the Godunov flux:Consistency: for all a ∈ [0,R], G(a,a) = f (a);Monotonicity and Lipschitz continuity:∃L > 0 : ∀(a,b) ∈ [0,R]2 there holds

0 ≤ ∂aG(a,b) ≤ L, −L ≤ ∂bG(a,b) ≤ 0.

Bardos-LeRoux-Nédélec boundary conditions

Consider the initial and boundary value problemut + f (u)x = 0, for t > 0, x < 0u(t ,0−) ≈ ub(t),u(0, x) = u0(x),

u is an entropy solution for the IBVP ifu is a Kruzhkov entropy solution in the interior of R+ × R−,u satisfies the boundary condition in the (BLN) sense

for a.e. t > 0 ∀k ∈ convu(t ,0−),ub(t)q(u(t ,0−), k) ≡ sign(u(t ,0−)−k)

(f (u(t ,0−))− f (k)

)≥ 0 (∗)

A reformulation of BLN:

u satisfies BLN (∗) ⇐⇒ f (u(t ,0−)) = G(u(t ,0−),ub(t)).

(known since [Dubois, LeFloch JDE’88])

The junction as a family of IBVPs

Fix ~ρ0 = (ρ10, . . . , ρ

m+n0 ) s.t. ρh

0 ∈ L∞(Ωh, [0,R]), ∀h ∈ 1, . . . ,m + n.We look for ~ρ = (ρ1, . . . , ρm+n) s.t.∀h, ρh ∈ L∞(R+ × Ωh, [0,R]) is an entropy solution of

ρh,t + fh(ρh)x = 0, on R+ × Ωh,

ρh(t ,0) ≈ vh(t), on R+,

ρh(0, x) = ρh0(x), on Ωh,

where ~v = (v1, . . . , vm+n) : R+ → [0,R]m+n is to be chosendepending on the JCC one wants to express,in particular, ensuring conservation at the junction.

The case of VV limits: require that vh be the same on all roads

[A., Cancès JHDE’15],[A., Mitrovic AnnIHP’15]:motivations in the “discontinuous-flux” case m = n = 1.[A., Cancès CompGS’13, JHDE’15]:examples of different admissibility criteria (e.g. vanishing capillarity).

The junction as a family of IBVPs

Fix ~ρ0 = (ρ10, . . . , ρ

m+n0 ) s.t. ρh

0 ∈ L∞(Ωh, [0,R]), ∀h ∈ 1, . . . ,m + n.We look for ~ρ = (ρ1, . . . , ρm+n) s.t.∀h, ρh ∈ L∞(R+ × Ωh, [0,R]) is an entropy solution of

ρh,t + fh(ρh)x = 0, on R+ × Ωh,

ρh(t ,0) ≈ vh(t), on R+,

where ~v = (v1, . . . , vm+n) : R+ → [0,R]m+n is to be chosendepending on the JCC one wants to express,in particular, ensuring conservation at the junction.

The case of VV limits: require that vh be the same on all roads

[A., Cancès JHDE’15],[A., Mitrovic AnnIHP’15]:motivations in the “discontinuous-flux” case m = n = 1.[A., Cancès CompGS’13, JHDE’15]:examples of different admissibility criteria (e.g. vanishing capillarity).

Admissibility at the junction

Continuity of ρ at the junction - up to boundary layers! - is desired.

Definition I (Formalized from heuristics of the model)

Given ~ρ0 i.c., we say that ~ρ = (ρ1, . . . , ρm+n) is an admissible solutionfor the Cauchy problem on the network, if ∃p in L∞(R+, [0,R]) s.t.

each component ρh is entropy solution for the IBVPρh,t + fh(ρh)x = 0, on R+ × Ωh,

ρh(t ,0) ≈ p(t), on R+,

(this includes the “density-at-junction” condition ∀h vh(t) = p(t))and “flux-at-junction” condition (conservativity) holds, i.e.∑m

i=1G(ρi (t ,0−),p(t)) =

∑m+n

j=m+1G(p(t), ρj (t ,0+)), for a.e. t .

Drawback: with this definition

uniqueness is not obvious...existence is not obvious.

ρh(t ,0) ≈ p(t), on R+,

i=1G(ρi (t ,0−),p(t)) =

∑m+n

j=m+1G(p(t), ρj (t ,0+)), for a.e. t .

ρh(t ,0) ≈ p(t), on R+,

i=1G(ρi (t ,0−),p(t)) =

∑m+n

j=m+1G(p(t), ρj (t ,0+)), for a.e. t .

The vanishing viscosity germ

JCC⇔ “Germ” ∼ stationary road-wise constant admissible sol.

The germ underlying the above description of admissibility is

~u = (u1, . . . ,um+n) : ∃p ∈ [0,R] s.t. :∑m

i=1Gi (ui ,p) =

∑m+n

j=m+1Gj (p,uj )

Gi (ui ,p) = fi (ui ), Gj (p,uj ) = fj (uj ) ∀i , j

Proposition (the Riemann solver RVV at junction)

Given any ~u = (u1, . . . ,um+n) ∈ [0,R]m+n the corresponding Riemannproblem at the junction has an admissible solution ~ρ = RVV [~u].The vector of traces ~γρ = (ρ1(0−), . . . , ρm+n(0+)) belongs to GVV .

Idea of proof: given Riemann data ~u, construct p(the common boundary-value in the def. of admissibility) by solving

find p~u s.t.∑m

i=1Gi (ui ,p~u) =

∑m+n

j=m+1Gj (p~u,uj )

~u = (u1, . . . ,um+n) : ∃p ∈ [0,R] s.t. :∑m

i=1Gi (ui ,p) =

∑m+n

j=m+1Gj (p,uj )

find p~u s.t.∑m

i=1Gi (ui ,p~u) =

∑m+n

j=m+1Gj (p~u,uj )

~u = (u1, . . . ,um+n) : ∃p ∈ [0,R] s.t. :∑m

i=1Gi (ui ,p) =

∑m+n

j=m+1Gj (p,uj )

find p~u s.t.∑m

i=1Gi (ui ,p~u) =

∑m+n

j=m+1Gj (p~u,uj )

Example

Consider a junction consisting of two incoming and one outgoing roads.

f1(x) = −x2 + 1, f2(x) = −2x2 + 2, f3(x) = −3x2 + 3.

Given the initial condition (ρ10 = −

√1/2, ρ2

0 = 1/4, ρ30 =

√1/6) one can trace

p 7→ Gi(ρi0, p), i = 1, 2, p 7→ G3(p, ρ3

For all p ∈ [−√

1/6, 0],

2∑i=1

Gi(ρi0, p) = G3(p, ρ3

0) = 2.5.

The fluxes G1,2(ρi0, p), G3(p, ρ3

0) areindependent of p ∈ [−

√1/6, 0].

NB: In practice, p~u can be found by regula falsi method.

Germ-based equivalent definitions of admissibility

A function ~ρ = (ρ1, . . . , ρm+n) is an admissible solution if and only if

Definition II (trace-based: used to prove uniqueness)

∀h ∈ 1, . . . ,m + n, ρh is a Kruzhkov solution on the road Ωh;traces-in-germ condition holds:

for a.e. t ∈ R+, ~γρ(t) := (ρ1(t ,0−), . . . , ρm+n(t ,0+)) ∈ GVV .

cf. [Garavello, Natalini, Piccoli, Terracina ’07](admissibility in terms of Riemann solver at junction)

Definition III (integral formulation: used to prove existence)

∀h ∈ 1, . . . ,m + n, ρh is a Kruzhkov solution on the road Ωh;adapted entropy inequalities hold: ∀ξ ∈ D(R+ × R), ξ ≥ 0

∀~k ∈ GVV

∑m+n

(∫R+

∫Ωh

|ρh − kh|ξt + qh(ρh, kh)ξx dx dt)≥ 0.

cf. [Baiti, Jenssen’97],[Audusse, Perthame’05]. These are Kato ineq.!

Germ-based equivalent definitions of admissibility

A function ~ρ = (ρ1, . . . , ρm+n) is an admissible solution if and only if

Definition II (trace-based: used to prove uniqueness)

∀h ∈ 1, . . . ,m + n, ρh is a Kruzhkov solution on the road Ωh;traces-in-germ condition holds:

for a.e. t ∈ R+, ~γρ(t) := (ρ1(t ,0−), . . . , ρm+n(t ,0+)) ∈ GVV .

cf. [Garavello, Natalini, Piccoli, Terracina ’07](admissibility in terms of Riemann solver at junction)

Definition III (integral formulation: used to prove existence)

∀h ∈ 1, . . . ,m + n, ρh is a Kruzhkov solution on the road Ωh;adapted entropy inequalities hold: ∀ξ ∈ D(R+ × R), ξ ≥ 0

∀~k ∈ GVV

∑m+n

(∫R+

∫Ωh

|ρh − kh|ξt + qh(ρh, kh)ξx dx dt)≥ 0.

cf. [Baiti, Jenssen’97],[Audusse, Perthame’05]. These are Kato ineq.!

Crucial properties of GVV

The germ GVV is “complete”: namely,the Riemann solver RVV is defined for all data.The germ GVV is “dissipative”: namely, for all ~k ,~c ∈ GVV∑m

i=1qi (ki , ci )−

∑m+n

j=m+1qj (kj , cj ) ≥ 0. (#)

The germ GVV is “maximal”: namely,if ~k fulfills (#) for all ~c ∈ GVV , then ~k ∈ GVV .

Lemma (Oleinik-like condition)

GVV is characterized by a “graph above-graph below” condition (cf.[Diehl JHDE’09] in the n = m = 1 case) + conservativity condition.

The maximality can be refined: considerGo

VV = strict “graph above-graph below” condition.The subset Go

VV of GVV is “definite”: namely,

taking ~c ∈ GoVV is enough to deduce from (#) that ~k ∈ GVV .

i=1qi (ki , ci )−

∑m+n

j=m+1qj (kj , cj ) ≥ 0. (#)

i=1qi (ki , ci )−

∑m+n

j=m+1qj (kj , cj ) ≥ 0. (#)

i=1qi (ki , ci )−

∑m+n

j=m+1qj (kj , cj ) ≥ 0. (#)

i=1qi (ki , ci )−

∑m+n

j=m+1qj (kj , cj ) ≥ 0. (#)

Well-posedness in the frame of admissible solutions

Theorem (Main result)

There exists an admissible solution for all L∞ datum.Moreover, such solutions are VV limits.If ~ρ and ~ρ are admissible solutions corresponding to ~ρ0 and ~ρ0,∑m+n

h=1‖ρh(t)− ρh(t)‖L1(Ωh) ≤

∑m+n

h=1‖ρ0

h − ρ0h‖L1(Ωh).

(L1-contractivity, monotonicity). Also Kato inequality holds.=⇒ uniqueness of an admissible solution to Cauchy problem.

Proof of uniqueness: Kruzhkov-per-road (doubling of variables) +existence of junction traces γ~ρ, γ~ρ =⇒ up-to-junction Kato inequality:

−∫R+

∫Ωh

(|ρh − ρh|ξt + qh(ρh, ρh)ξx

)≤ RHS[γ~ρ, γ~ρ] ∀ξ ≥ 0

By Def.II, γ~ρ, γ~ρ ∈ GVV . Then dissipativity (#) =⇒ RHS[γ~ρ, γ~ρ] ≤ 0.

∑m+n

h=1‖ρ0

−∫R+

∫Ωh

)≤ RHS[γ~ρ, γ~ρ] ∀ξ ≥ 0

∑m+n

h=1‖ρ0

−∫R+

∫Ωh

)≤ RHS[γ~ρ, γ~ρ] ∀ξ ≥ 0

Existence of admissible solutions

Proof of existence:Recall some of [Coclite, Garavello SIAM’10] results:

(subseq. of) VV approximations ~ρ ε converges a.e. to a limit ~ρ ;for all ε > 0, the semigroup of VV approximations isorder-preserving / L1-contractive / fulfills Kato inequality.

Standard theory: each component ρh of ~ρ, s.t. ρεh → ρh,is a Kruzhkov entropy solution in Ωh.All ~k ∈ Go

VV can be obtained as VV limits.Tool: explicit construction of viscosity profiles ~k ε

(based upon the Oleinik “graph above-graph below” condition).Pass to the limit ε→ 0 in Kato ineq. written for ~ρ ε and ~k ε.We get adapted entropy inequalities ⇒ ρ fulfills Def. III.

Alternative proof: (for monotone junction Riemann solvers)use Godunov numerical scheme to construct solutions

Godunov scheme is well-balanced:~k in the germ (stationary solutions) are exact discrete solutionsmonotonicity of Riemann solver⇒ discrete Kato inequalities

follow the same steps of passage to the limit.

G r a z i e !Thank you for your attention!

well-posedness for some lwr models on a junctionacesar/slidesanconet/andreia...introductionmonotone...

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