Solutions for a hyperbolic model of multiphase flow and related - - PowerPoint PPT Presentation

Solutions for a hyperbolic model of multiphase flow and related numerical issues

Debora Amadori

University of L’Aquila (Italy)

AMIS2012, June 20 2012

Outline

Outline

1

Part I: A multiphase flow model Description Basics on the homogeneous system Comparison with other models Solutions for the homogeneous system The algorithm The system with a reaction term

2

Part II: Error estimates for scalar laws with source The setting Fractional Step/ Well Balanced at a glance L1 stability result

Part I: A multiphase flow model Description

A 1–D model for multiphase reactive flow

     vt − ux = 0 ut + p(v, λ)x = 0 λt = α τ (p − pe)λ(λ − 1) v > 0: specific volume, u: velocity, λ: mass density fraction of vapor in the fluid 0 ≤ λ ≤ 1; λ = 0 : liquid; λ = 1 : vapor;

Part I: A multiphase flow model Description

A 1–D model for multiphase reactive flow

     vt − ux = 0 ut + p(v, λ)x = 0 λt = α τ (p − pe)λ(λ − 1) v > 0: specific volume, u: velocity, λ: mass density fraction of vapor in the fluid 0 ≤ λ ≤ 1; λ = 0 : liquid; λ = 1 : vapor; α > 0, pe a fixed equilibrium pressure, τ a reaction time p: pressure p = p(v, λ) = A(λ) vγ , A(λ) > 0 , A′(λ) > 0

[Fan, SIAM J. Appl. Math. (2000)]

Part I: A multiphase flow model Description

✲ v ✻ p λ = 0 λ = 1 pe . . . . . . . . vm . . . . . . . . vM metastable vapor ✏ ✏ ✏ ✮ stable vapor ✘ ✘ ✾ stable liquid P P q metastable liquid ✲ The system shows all major wave patterns observed in shock tube experiments on retrograde fluids.

[Thompson, Carofano, Kim, 1986; Thompson, Chaves, Meier, Kim and Speckmann, 1987].

The model may take into account viscosity terms      vt − ux = 0 ut + px = εuxx λt = α τ (p − pe)λ(λ − 1) + bελxx Riemann problem for the 0-viscosity and 0-relaxation limit: [Corli and Fan (2005)].

Part I: A multiphase flow model Description

Aim of this work

For a fixed relaxation time τ > 0, look for global (in time) solutions of the Cauchy problem with large BV data (v, u, λ)(0, x) =

• vo(x), uo(x), λo(x)
• ,

In the spirit of the fractional step method [Dafermos& Hsiao, 1982], the analysis will consider

the homogeneous system: the 3 × 3 system of conservation laws (H) 8 < : vt − ux = 0 ut + p(v, λ)x = 0 λt = 0 the complete system, where the source term is added by time discretization.

Part I: A multiphase flow model Description

Aim of this work

For a fixed relaxation time τ > 0, look for global (in time) solutions of the Cauchy problem with large BV data (v, u, λ)(0, x) =

• vo(x), uo(x), λo(x)
• ,

In the spirit of the fractional step method [Dafermos& Hsiao, 1982], the analysis will consider

the homogeneous system: the 3 × 3 system of conservation laws (H) 8 < : vt − ux = 0 ut + p(v, λ)x = 0 λt = 0 the complete system, where the source term is added by time discretization.

The relaxation limit τ → 0.

[Joint work with: Andrea Corli (University of Ferrara)]

Part I: A multiphase flow model Basics on the homogeneous system

The homogeneous system

Assume γ = 1. The 3 × 3 system (H)    vt − ux = 0 ut + (A(λ)/v)x = 0 λt = 0 is strictly hyperbolic: e1,3 = ±√−pv = ±

• A(λ)/v,

e2 = 0 . Two fields are GNL, one is LD.

Part I: A multiphase flow model Basics on the homogeneous system

The homogeneous system

Assume γ = 1. The 3 × 3 system (H)    vt − ux = 0 ut + (A(λ)/v)x = 0 λt = 0 is strictly hyperbolic: e1,3 = ±√−pv = ±

• A(λ)/v,

e2 = 0 . Two fields are GNL, one is LD.

• From the third equation:

λ = λo(x) ⇒ 2 × 2 system with non-homogeneous flux, possibly discontinuous.

Part I: A multiphase flow model Basics on the homogeneous system

The homogeneous system

Assume γ = 1. The 3 × 3 system (H)    vt − ux = 0 ut + (A(λ)/v)x = 0 λt = 0 is strictly hyperbolic: e1,3 = ±√−pv = ±

• A(λ)/v,

e2 = 0 . Two fields are GNL, one is LD.

• From the third equation:

λ = λo(x) ⇒ 2 × 2 system with non-homogeneous flux, possibly discontinuous.

• Special case:

λo(x) =

• λℓ

x < 0 λr x > 0

Part I: A multiphase flow model Basics on the homogeneous system

The homogeneous system

Assume γ = 1. The 3 × 3 system (H)    vt − ux = 0 ut + (A(λ)/v)x = 0 λt = 0 is strictly hyperbolic: e1,3 = ±√−pv = ±

• A(λ)/v,

e2 = 0 . Two fields are GNL, one is LD.

• From the third equation:

λ = λo(x) ⇒ 2 × 2 system with non-homogeneous flux, possibly discontinuous.

• Special case:

λo(x) =

• λℓ

x < 0 λr x > 0 Look for weak BV solutions, globally defined in time, with possibly large data

Part I: A multiphase flow model Comparison with other models

Comparison with other models

In Eulerian coordinates, for p = A(λ)ρ and ρ = 1/v, the system (H) rewrites as    ρt + (ρu)x = 0, (ρu)t +

• ρu2 + p(ρ, λ)
• x

= 0, (ρλ)t + (ρλu)x = 0 . In [Benzoni-Gavage, 1991] many models for diphasic flows are proposed, for instance    (ρlRl)t + (ρlRlul)x = 0 (ρgRg)t + (ρgRgug)x = 0 (ρlRlul + ρgRgug)t + (ρlRlu2

l + ρgRgu2 g + p)x

= 0 . Here l and g stand for liquid and gas; ρl, Rl, ul are the liquid density, phase fraction, velocity, and analogously for the gas, Rl + Rg = 1, p = Aρg.

Part I: A multiphase flow model Comparison with other models

Comparison with other models

If ul = ug, ρl = 1 and Rl > 0, define the concentration c = ρgRg ρlRl then [Peng, 1994]    (Rl)t + (Rlu)x = 0 (Rlc)t + (Rlcu)x = 0 (Rl(1 + c)u)t +

• Rl(1 + c)u2 + p
• x

= 0 (P) with p = Ac Rl 1 − Rl . System (P) is strictly hyperbolic for c > 0; the eigenvalues coincide with u at c = 0. If c ≡ 0 then (P) reduces to the pressureless gasdynamics system.

Part I: A multiphase flow model Comparison with other models

Comparison with other models

If ul = ug, ρl = 1 and Rl > 0, define the concentration c = ρgRg ρlRl then [Peng, 1994]    (Rl)t + (Rlu)x = 0 (Rlc)t + (Rlcu)x = 0 (Rl(1 + c)u)t +

• Rl(1 + c)u2 + p
• x

= 0 (P) with p = Ac Rl 1 − Rl . System (P) is strictly hyperbolic for c > 0; the eigenvalues coincide with u at c = 0. If c ≡ 0 then (P) reduces to the pressureless gasdynamics system. System (P) is analogous to (H), with λ = c 1 + c , but the pressure laws differ when λ , c ∼ 0.

Part I: A multiphase flow model Comparison with other models

The Riemann problem

The Cauchy problem for      vt − ux = 0 ut +

• A(λ)

v

• x

= 0 λt = 0 (v, u, λ)(0, x) = (vℓ, uℓ, λℓ) x < 0 (vr, ur, λr) x > 0 can be solved for any pair of initial data (with vℓ, vr > 0 and λℓ, λr ∈ [0, 1]). ε2 ✁ ✁✁ ε3 ❅ ❅ ❅ ε1 Phase waves are stationary: p, u are conserved across them (“kinetic condition”). pℓ = pr uℓ = ur

Part I: A multiphase flow model Comparison with other models

Known results

For small BV data: well-posedness of the Cauchy problem.

[Glimm 1965; Bressan, Hyperbolic systems..., 2000.]

If λ is constant: the Cauchy problem for vt − ux = 0, ut + (A/v)x = 0 has a global solution if only Tot.Var. (vo, uo) < ∞

[Nishida (1968)]

If p(v) = A vγ , with γ > 1 global existence if (γ − 1)Tot.Var. (uo, vo) is small

[Nishida-Smoller, DiPerna 1973].

If λ is not constant, [Peng (1994)]; results by compensated compactness: [B´

ereux, Bonnetier, & LeFloch (1997); Gosse (2001); Lu (2003)], but for different pressure laws.

For the full Euler system: [Liu (1977), Temple (1981)].

Part I: A multiphase flow model Solutions for the homogeneous system

Solutions for the homogeneous system

Assume: vo(x) ≥ v > 0 , a = √ A , λo(x) ∈ [0, 1] and define Ao = 2 sup

n

• j=1

|a(λ(xj)) − a(λ(xj−1))| a(λ(xj)) + a(λ(xj−1)) . where the supremum is taken over all n ≥ 1, xo < x1 < . . . < xn.

Part I: A multiphase flow model Solutions for the homogeneous system

Solutions for the homogeneous system

Assume: vo(x) ≥ v > 0 , a = √ A , λo(x) ∈ [0, 1] and define Ao = 2 sup

n

• j=1

|a(λ(xj)) − a(λ(xj−1))| a(λ(xj)) + a(λ(xj−1)) . where the supremum is taken over all n ≥ 1, xo < x1 < . . . < xn. Observe that Ao ∼ Tot.Var.

• log
• a(λo)
• .

Part I: A multiphase flow model Solutions for the homogeneous system

Solutions for the homogeneous system

Assume: vo(x) ≥ v > 0 , a = √ A , λo(x) ∈ [0, 1] and define Ao = 2 sup

n

• j=1

|a(λ(xj)) − a(λ(xj−1))| a(λ(xj)) + a(λ(xj−1)) . where the supremum is taken over all n ≥ 1, xo < x1 < . . . < xn. Observe that Ao ∼ Tot.Var.

• log
• a(λo)
• .

Theorem 1: For a suitable decreasing function H : (0, 1/2] → [0, ∞), if Ao < 1 2 , Tot.Var. (log po) + 1 inf ao Tot.Var. uo < H(Ao) , then the Cauchy problem for (H) has a weak entropic solution (v, u, λ) defined for t ≥ 0, with uniformly bounded total variation.

Part I: A multiphase flow model Solutions for the homogeneous system

Solutions for the homogeneous system

Assume: vo(x) ≥ v > 0 , a = √ A , λo(x) ∈ [0, 1] and define Ao = 2 sup

n

• j=1

|a(λ(xj)) − a(λ(xj−1))| a(λ(xj)) + a(λ(xj−1)) . where the supremum is taken over all n ≥ 1, xo < x1 < . . . < xn. Observe that Ao ∼ Tot.Var.

• log
• a(λo)
• .

Theorem 1: For a suitable decreasing function H : (0, 1/2] → [0, ∞), if Ao < 1 2 , Tot.Var. (log po) + 1 inf ao Tot.Var. uo < H(Ao) , then the Cauchy problem for (H) has a weak entropic solution (v, u, λ) defined for t ≥ 0, with uniformly bounded total variation.

[Amadori, Corli, SIAM J. Math. Anal., 2008]

Part I: A multiphase flow model The algorithm

The algorithm

Use a suitable version of a wave-front tracking algorithm (mainly from [Bressan

(2000)] and [Amadori-Guerra (2001)]):

approximate the initial data with piecewise constant functions; solve the interactions of sonic waves (families 1 and 3) and the interactions with the stationary phase waves (family 2) ✡ ✡ ✡ δ3 ❆ ❆ ❆ δ1 ✁ ✁ ε3 ❅ ❅ ε1

• α3

✡ ✡ ✡β3 ✁ ✁ ε3 ❅ ❅ ε1 δ2 ✁ ✁ ✁ δ3

• ε3

❆ ❆ ε1

• δ3

✘✘✘ ✘ n p

• δ3

Part I: A multiphase flow model The algorithm

The algorithm

Use a suitable version of a wave-front tracking algorithm (mainly from [Bressan

(2000)] and [Amadori-Guerra (2001)]):

approximate the initial data with piecewise constant functions; solve the interactions of sonic waves (families 1 and 3) and the interactions with the stationary phase waves (family 2) ✡ ✡ ✡ δ3 ❆ ❆ ❆ δ1 ✁ ✁ ε3 ❅ ❅ ε1

• α3

✡ ✡ ✡β3 ✁ ✁ ε3 ❅ ❅ ε1 δ2 ✁ ✁ ✁ δ3

• ε3

❆ ❆ ε1

• δ3

✘✘✘ ✘ n p

• δ3

Show that the algorithm is well defined Control the total variation of the approx. solution Pass to the limit by compactness.

Part I: A multiphase flow model The system with a reaction term

The system with a reaction term

We come back to the complete system: (S)      vt − ux = 0 ut + p(v, λ)x = 0 λt = 1 τ g(p, λ) , g = (p − pe)λ(λ − 1)

Part I: A multiphase flow model The system with a reaction term

The system with a reaction term

We come back to the complete system: (S)      vt − ux = 0 ut + p(v, λ)x = 0 λt = 1 τ g(p, λ) , g = (p − pe)λ(λ − 1) Equilibrium points of the source term: λ = 0, liquid phase: the system reduces to vt − ux = 0, ut + p(v, 0)x = 0 . λ = 1, vapor phase Equilibrium pressure. Here the system reduces to p = pe , u = const. , λ = λ(x) .

Part I: A multiphase flow model The system with a reaction term

Known results

About global existence in time for hyperbolic systems of conservation laws with source terms:

[T.-P. Liu, CMP (1979)]; [Dafermos & Hsiao Indiana J. (1982)]

About p-system with γ = 1 and source term: [Luskin-Temple, 1982]; [Poupaud, Rascle,

Vila 1995] for the isothermal Euler-Poisson system with large data. [Luo-Natalini-Yang 2000, Amadori-Guerra, 2001]:

global existence of BV solutions for vt − ux = 0 ut + (1/v)x = 1

τ r(v, u) ,

and relaxation limit for τ → 0. Tipical case: r(v, u) = a(v) − u .

Part I: A multiphase flow model The system with a reaction term

The case λ ∼ 0

For τ > 0 fixed, we focus on the case λ ∼ 0. From the equation λt = 1 τ (p − pe)λ(λ − 1) , the sign of (p − pe) determines the behavior of the equation for λ: provided that p − pe ≥ c > 0 , λ ≤ µ < 1 then we get λt ≤ − c τ (1 − µ)λ and the contribution of the source term decays exponentially in time.

[Amadori & Corli, Nonl. Anal., 2010]

Part I: A multiphase flow model The system with a reaction term

The case λ ∼ 0

Theorem 2: For τ > 0 fixed, assume suitable bounds on Tot.Var. po, Tot.Var. uo and that 0 < c ≤ po(x) − pe ≤ c ∀x . Then if λo∞ , Tot.Var. λo are sufficiently small, the Cauchy problem for the system (S) has a weak entropy solution (v, u, λ) defined for t ≥ 0. Furthermore, for suitable constants 0 < c1 < c1 and c2 > 0, one has c1 ≤ p(x, t) − pe ≤ c1 , 0 ≤ λ(x, t) ≤ λo∞ e−c2 t

τ

and the bounds on Tot.Var. (v, u, λ) (·, t) are independent of τ.

Part I: A multiphase flow model The system with a reaction term

Applying the fractional step scheme...

t = n∆t

• ❆

❆ ❆ 2 ❇ ❇ ❇

• ❆

❆ ❆ 1

• np

✄ ✄ ✄ ✂ ✂ ✂ ❆ ❆ ❆ 3

Part I: A multiphase flow model The system with a reaction term

Applying the fractional step scheme...

t = n∆t

• ❆

❆ ❆ 2 ❇ ❇ ❇

• ❆

❆ ❆ 1

• np

✄ ✄ ✄ ✂ ✂ ✂ ❆ ❆ ❆ 3 t → λ(t)∞ decays ⇒ control on Tot.Var. (p, u, λ) ⇒ control on p from below

Part I: A multiphase flow model The system with a reaction term

Applying the fractional step scheme...

t = n∆t

• ❆

❆ ❆ 2 ❇ ❇ ❇

• ❆

❆ ❆ 1

• np

✄ ✄ ✄ ✂ ✂ ✂ ❆ ❆ ❆ 3 t → λ(t)∞ decays ⇒ control on Tot.Var. (p, u, λ) ⇒ control on p from below For every t > s ≥ 0, one has λ(·, t) − λ(·, s)L1

loc ≤ Lτ (t − s + ∆t)

Lτ = C1 + C2 τ e− C3s

τ

.

Part I: A multiphase flow model The system with a reaction term

Applying the fractional step scheme...

t = n∆t

• ❆

❆ ❆ 2 ❇ ❇ ❇

• ❆

❆ ❆ 1

• np

✄ ✄ ✄ ✂ ✂ ✂ ❆ ❆ ❆ 3 t → λ(t)∞ decays ⇒ control on Tot.Var. (p, u, λ) ⇒ control on p from below For every t > s ≥ 0, one has λ(·, t) − λ(·, s)L1

loc ≤ Lτ (t − s + ∆t)

Lτ = C1 + C2 τ e− C3s

τ

. Notice: Lτ → C1 for s > 0, while Lτ → +∞ for s = 0.

Part I: A multiphase flow model The system with a reaction term

The relaxation limit

Theorem 3: For τ > 0, let the initial data (v, u, λ)(0, x) =

• vτ
• (x), uτ
• (x), λτ
• (x)
• ,

satisfy the bounds of Theorem 2 uniformly in τ. Assume that vτ

• → vo ,

uτ

• → uo

in L1

loc(R) ,

as τ → 0 . Let (vτ, uτ, λτ)(x, t) be the weak entropic solution provided by Theorem 2. Then there exists a (sub)sequence τn → 0 s. t.

Part I: A multiphase flow model The system with a reaction term

The relaxation limit

Theorem 3: For τ > 0, let the initial data (v, u, λ)(0, x) =

• vτ
• (x), uτ
• (x), λτ
• (x)
• ,

satisfy the bounds of Theorem 2 uniformly in τ. Assume that vτ

• → vo ,

uτ

• → uo

in L1

loc(R) ,

as τ → 0 . Let (vτ, uτ, λτ)(x, t) be the weak entropic solution provided by Theorem 2. Then there exists a (sub)sequence τn → 0 s. t. λτn → in L1

loc(R × (0, ∞))

(vτn, uτn) → ( v, u) in L1

loc(R × [0, ∞)) ,

where ( v, u) is a weak solution for t ≥ 0 to the Cauchy problem vt − ux = 0 ut + p(v, 0)x = 0 , v(x, 0) = vo(x) u(x, 0) = uo(x) .

Part I: A multiphase flow model The system with a reaction term

The relaxation limit: entropies

Entropy inequality for τ > 0: ηt + qx ≤ 1 τ ηλ · g(v, λ) . Dissipative if ηλ · g ≤ 0 .

Part I: A multiphase flow model The system with a reaction term

The relaxation limit: entropies

Entropy inequality for τ > 0: ηt + qx ≤ 1 τ ηλ · g(v, λ) . Dissipative if ηλ · g ≤ 0 . Entropy-entropy flux pair for the 3 × 3 system: η = u2 2 − A(λ) log v + φ(λ), q = A(λ)u v

Part I: A multiphase flow model The system with a reaction term

The relaxation limit: entropies

Entropy inequality for τ > 0: ηt + qx ≤ 1 τ ηλ · g(v, λ) . Dissipative if ηλ · g ≤ 0 . Entropy-entropy flux pair for the 3 × 3 system: η = u2 2 − A(λ) log v + φ(λ), q = A(λ)u v For a suitable choice of φ, the entropy is convex and dissipative w.r.t. the source term.

Part I: A multiphase flow model The system with a reaction term

The relaxation limit: entropies

Entropy inequality for τ > 0: ηt + qx ≤ 1 τ ηλ · g(v, λ) . Dissipative if ηλ · g ≤ 0 . Entropy-entropy flux pair for the 3 × 3 system: η = u2 2 − A(λ) log v + φ(λ), q = A(λ)u v For a suitable choice of φ, the entropy is convex and dissipative w.r.t. the source term. Hence, for τn → 0, the limit ( v, u) satisfies the entropy inequality for

• η = u2

2 − A(0) log v,

• q = A(0)u

v

Part I: A multiphase flow model The system with a reaction term

Conclusion (Part I)

(S)      vt − ux = 0 ut + p(v, λ)x = 0 λt = 1 τ (p − pe)λ(λ − 1) Done: λ ∼ 0, p > pe, for τ fixed; limit τ → 0. Similar for λ ∼ 1, p < pe Open: p ∼ pe, 0 < λ < 1

Part II: Error estimates for scalar laws with source

Part II: Error estimates for scalar laws with source

Part II: Error estimates for scalar laws with source The setting

Error estimates for scalar laws with source

[Joint work with Laurent Gosse (IAC-CNR, Rome)]

∂tu + ∂xf(u) = k(x)g(u) (∗) Initial data uo ∈ L1 ∩ BV (R), g smooth, k ∈ L1(R) Non resonance assumption: f ′ > 0

Part II: Error estimates for scalar laws with source The setting

Error estimates for scalar laws with source

[Joint work with Laurent Gosse (IAC-CNR, Rome)]

∂tu + ∂xf(u) = k(x)g(u) (∗) Initial data uo ∈ L1 ∩ BV (R), g smooth, k ∈ L1(R) Non resonance assumption: f ′ > 0 No dissipativity of the source: k(x)g′(u) ≤ 0 (otherwise, L1 distance between two solutions of (∗) decays in time)

Part II: Error estimates for scalar laws with source The setting

Error estimates for scalar laws with source

[Joint work with Laurent Gosse (IAC-CNR, Rome)]

∂tu + ∂xf(u) = k(x)g(u) (∗) Initial data uo ∈ L1 ∩ BV (R), g smooth, k ∈ L1(R) Non resonance assumption: f ′ > 0 No dissipativity of the source: k(x)g′(u) ≤ 0 (otherwise, L1 distance between two solutions of (∗) decays in time) Approximation (theoretical, numerical) of (∗), different approaches: Well Balanced (WB), Fractional Step (FS) Aim: compare WB approach vs FS. Useful technique: L1 stability theory for hyperbolic systems of conservation laws.

[Bressan, T.-P. Liu ]

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Fractional Step

Fix ∆t > 0, set tn = n∆t.

1

∂tu + ∂xf(u) = 0

• ver each (tn, tn+1)

2

u(tn+) = u(tn−) + ∆t k(·)g(u(tn−)) at t = tn

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Fractional Step

Fix ∆t > 0, set tn = n∆t.

1

∂tu + ∂xf(u) = 0

• ver each (tn, tn+1)

2

u(tn+) = u(tn−) + ∆t k(·)g(u(tn−)) at t = tn Error estimates for time splitting operators:

[Tang& Teng (1995); Langseth, A. Tveito & Winther (1996)]:

u∆t(t, ·) − u(t, ·)L1(R) ≤ C exp(sup[k(x)g′(u)]t) √ ∆t , ∀ t ∈ [0, T] It follows from application of Gronwall Lemma

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Fractional Step

Fix ∆t > 0, set tn = n∆t.

1

∂tu + ∂xf(u) = 0

• ver each (tn, tn+1)

2

u(tn+) = u(tn−) + ∆t k(·)g(u(tn−)) at t = tn Error estimates for time splitting operators:

[Tang& Teng (1995); Langseth, A. Tveito & Winther (1996)]:

u∆t(t, ·) − u(t, ·)L1(R) ≤ C exp(sup[k(x)g′(u)]t) √ ∆t , ∀ t ∈ [0, T] It follows from application of Gronwall Lemma Numerical proofs show that the exponential amplification can actually occur

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Well Balanced

Fix ∆x > 0, set xk = k∆x.

1

∂tu + ∂xf(u) = 0

• ver each (xk, xk+1)

2

f(u(t, xk+)) = f(u(t, xk−)) + xk+1

xk

k(x)g(u(t, x)) dx at x = xk (stationary wave located at xk)

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Well Balanced

Fix ∆x > 0, set xk = k∆x.

1

∂tu + ∂xf(u) = 0

• ver each (xk, xk+1)

2

f(u(t, xk+)) = f(u(t, xk−)) + xk+1

xk

k(x)g(u(t, x)) dx at x = xk (stationary wave located at xk) Good features: It preserves steady state solutions

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Well Balanced

Fix ∆x > 0, set xk = k∆x.

1

∂tu + ∂xf(u) = 0

• ver each (xk, xk+1)

2

f(u(t, xk+)) = f(u(t, xk−)) + xk+1

xk

k(x)g(u(t, x)) dx at x = xk (stationary wave located at xk) Good features: It preserves steady state solutions It can be recasted as a homogeneous 2 × 2 system: setting a′ = k ∂tu + ∂xf(u) − g(u) ∂xa = 0 , ∂ta = 0 .

• Char. speeds: 0, f ′(u).

Strictly hyperbolic if f ′ > 0. Riemann Invariants: a, w(a, u) = φ−1 (φ(u) − a), with φ′ = f ′/g.

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Well Balanced

Fix ∆x > 0, set xk = k∆x.

1

∂tu + ∂xf(u) = 0

• ver each (xk, xk+1)

2

f(u(t, xk+)) = f(u(t, xk−)) + xk+1

xk

k(x)g(u(t, x)) dx at x = xk (stationary wave located at xk) Good features: It preserves steady state solutions It can be recasted as a homogeneous 2 × 2 system: setting a′ = k ∂tu + ∂xf(u) − g(u) ∂xa = 0 , ∂ta = 0 .

• Char. speeds: 0, f ′(u).

Strictly hyperbolic if f ′ > 0. Riemann Invariants: a, w(a, u) = φ−1 (φ(u) − a), with φ′ = f ′/g. It is non-conservative.

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

Well Balanced

Fix ∆x > 0, set xk = k∆x.

1

∂tu + ∂xf(u) = 0

• ver each (xk, xk+1)

2

f(u(t, xk+)) = f(u(t, xk−)) + xk+1

xk

k(x)g(u(t, x)) dx at x = xk (stationary wave located at xk) Good features: It preserves steady state solutions It can be recasted as a homogeneous 2 × 2 system: setting a′ = k ∂tu + ∂xf(u) − g(u) ∂xa = 0 , ∂ta = 0 .

• Char. speeds: 0, f ′(u).

Strictly hyperbolic if f ′ > 0. Riemann Invariants: a, w(a, u) = φ−1 (φ(u) − a), with φ′ = f ′/g. It is non-conservative. Wave-front tracking approximations (WFT) [Dafermos (1972)]

Part II: Error estimates for scalar laws with source Fractional Step/ Well Balanced at a glance

A numerical example

Rarefaction wave for ∂tu + ∂x(u2/2) = 0.2 u, u0(x) =

• x < 0

1 x > 0 (Heaviside)

Part II: Error estimates for scalar laws with source L1 stability result

L1 stability [Guerra, JDE (2004)]

Given ∂tv + ∂xf(v) = b′(x)g(v) , U1 = (b, v) ∂tu + ∂xf(u) = a′(x)g(u) , U2 = (a, u) let U1, U2 be WFT approximations with error parameter δ1, δ2.

Part II: Error estimates for scalar laws with source L1 stability result

L1 stability [Guerra, JDE (2004)]

Given ∂tv + ∂xf(v) = b′(x)g(v) , U1 = (b, v) ∂tu + ∂xf(u) = a′(x)g(u) , U2 = (a, u) let U1, U2 be WFT approximations with error parameter δ1, δ2. There exists a functional Φ(U1, U2)(t), equivalent to the L1 distance U1 −U2L1, that is almost decaying: d dtΦ(U1, U2)(t) ≤ C exp(κ TV (a))[δ1 + δ2] . The constants do not depend on time.

Part II: Error estimates for scalar laws with source L1 stability result

L1 stability [Guerra, JDE (2004)]

Given ∂tv + ∂xf(v) = b′(x)g(v) , U1 = (b, v) ∂tu + ∂xf(u) = a′(x)g(u) , U2 = (a, u) let U1, U2 be WFT approximations with error parameter δ1, δ2. There exists a functional Φ(U1, U2)(t), equivalent to the L1 distance U1 −U2L1, that is almost decaying: d dtΦ(U1, U2)(t) ≤ C exp(κ TV (a))[δ1 + δ2] . The constants do not depend on time. No need of Gronwall Lemma!

Part II: Error estimates for scalar laws with source L1 stability result

Error estimates for Wave-Front Tracking

Sending δ2 → 0, we recover u as a weak entropy solution. Let U1 = (v, b) be a WFT approximation of (u, a) of parameter δ. Then: u(t) − v(t)L1 ≤ eκT V {a} [u0 − v(0)L1 + C′a − bL1 + C δ t ]

Part II: Error estimates for scalar laws with source L1 stability result

Error estimates for Wave-Front Tracking

Sending δ2 → 0, we recover u as a weak entropy solution. Let U1 = (v, b) be a WFT approximation of (u, a) of parameter δ. Then: u(t) − v(t)L1 ≤ eκT V {a} [u0 − v(0)L1 + C′a − bL1 + C δ t ] One can choose v(0), b such that: u0 − v(0)L1 ≤ δ Tot.Var. {u0} , a − bL1 ≤ δ Tot.Var. {a} .

Part II: Error estimates for scalar laws with source L1 stability result

Error estimates for Wave-Front Tracking

Sending δ2 → 0, we recover u as a weak entropy solution. Let U1 = (v, b) be a WFT approximation of (u, a) of parameter δ. Then: u(t) − v(t)L1 ≤ eκT V {a} [u0 − v(0)L1 + C′a − bL1 + C δ t ] One can choose v(0), b such that: u0 − v(0)L1 ≤ δ Tot.Var. {u0} , a − bL1 ≤ δ Tot.Var. {a} . ⇒ u(t) − v(t)L1 ≤ δ eκT V {a} [Tot.Var. {u0} + C′TV {a} + C t ] . The L1 error is linear both in δ and in t! Again, the constants do not depend on time; they depend on the TV of the Riemann invariants.

Part II: Error estimates for scalar laws with source L1 stability result

Current work Extend the analysis to the Well-Balanced Godunov scheme. Possible extension to systems with source term ([Amadori, Gosse, Guerra (2002)]: L1 stability)

Part II: Error estimates for scalar laws with source L1 stability result

Current work Extend the analysis to the Well-Balanced Godunov scheme. Possible extension to systems with source term ([Amadori, Gosse, Guerra (2002)]: L1 stability)

Thank you!

Part II: Error estimates for scalar laws with source L1 stability result

The functional

Let U1 = (b, v) and U2 = (a, u) be two wave-front tracking approximations. Let z(t, x) be the Riemann coordinate related to U1 = (b, v).

Part II: Error estimates for scalar laws with source L1 stability result

The functional

Let U1 = (b, v) and U2 = (a, u) be two wave-front tracking approximations. Let z(t, x) be the Riemann coordinate related to U1 = (b, v). Weight functions W1(t, x) = κ1TV {z; (−∞, x)} W2(x) = exp

• κ2

∞

x

|a′(y)| dy

• κ1, κ2 to be determined.

Part II: Error estimates for scalar laws with source L1 stability result

The functional

Let U1 = (b, v) and U2 = (a, u) be two wave-front tracking approximations. Let z(t, x) be the Riemann coordinate related to U1 = (b, v). Weight functions W1(t, x) = κ1TV {z; (−∞, x)} W2(x) = exp

• κ2

∞

x

|a′(y)| dy

• κ1, κ2 to be determined.

Functional: Φ(U1(t), U2(t)) = x2

x1+Lt

W2(x) [W1(t, x)|p(x)| + |q(t, x)|] dx , where p = a − b, q(t, x) = u(t, x) − ω(t, x) with ω(t, x) = φ(a(x); b(x), v(t, x)) L = max

(a,u)∈K f ′(u) .