Riemann Problem for Shallow Water Equations with Porosity Stelian - - PowerPoint PPT Presentation

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

Riemann Problem for Shallow Water Equations with Porosity

Stelian ION∗, Dorin MARINESCU∗, S ¸tefan-Gicu CRUCEANU∗

∗”Gh. Mihoc - C. Iacob” Institute of Mathematical Statistics and

Applied Mathematics of ROMANIAN ACADEMY

The 36th “Caius Iacob” Conference on Fluid Mechanics and its Technical Applications October 29-30, 2015, Bucharest, Romania

Partially Supported by ANCS, CNDI - UEFISCDI PNII programme, 50/2012

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

Table of Contents

1 1. Introduction

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

2 2. Shock and Rarefaction Waves

2.1. Shock Waves 2.2. Rarefaction Waves

3 3. Dam Break Problem

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

4 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

The System of Equations and Riemann Problem

1-D Shallow Water Equations ∂t(θh) + ∂x(θhu) = 0, ∂t(θhu) + ∂x(θhu2) + θhg∂x(z + h) = 0. Significance of the variables: h - water depth; u - water speed; g - gravitational acceleration; z - altitude to the soil surface; θ - porosity of the plant cover.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

The System of Equations and Riemann Problem

Riemann Problem Solve ∂t(θh) + ∂x(θhu) = 0, ∂t(θhu) + ∂x(θhu2) + θhg∂x(h) = 0, ∂t(θ) = 0, with initial condition (θ, h, u) =

• (θL, hL, uL),

x < 0, (θR, hR, uR), x > 0.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

The System of Equations and Riemann Problem

Riemann Problem Solve ∂t(θh) + ∂x(θhu) = 0, ∂t(θhu) + ∂x(θhu2) + θhg∂x(h) = 0, ∂t(θ) = 0, with initial condition (θ, h, u) =

• (θL, hL, uL),

x < 0, (θR, hR, uR), x > 0.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Conservative hyperbolic systems. Weak Solutions.

Let w : R × [0, ∞) → Ω =

• Ω ⊂ Rp be the unknown field, and

f : Ω → Rp smooth (flux-function). Conservative hyperbolic systems ∂tw + ∂xf (w) = 0, w(x, 0) = w0(x).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Conservative hyperbolic systems. Weak Solutions.

Let w : R × [0, ∞) → Ω =

• Ω ⊂ Rp be the unknown field, and

f : Ω → Rp smooth (flux-function). Conservative hyperbolic systems ∂tw + ∂xf (w) = 0, w(x, 0) = w0(x). Nonlinear hyperbolic problems: notion of classical solution;

• ccurrence of discontinuities in the solution;

need: new def. of a sol. w not necessarily differentiable;

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Conservative hyperbolic systems. Weak Solutions.

Let w : R × [0, ∞) → Ω =

• Ω ⊂ Rp be the unknown field, and

f : Ω → Rp smooth (flux-function). Conservative hyperbolic systems ∂tw + ∂xf (w) = 0, w(x, 0) = w0(x). Nonlinear hyperbolic problems: notion of classical solution;

• ccurrence of discontinuities in the solution;

need: new def. of a sol. w not necessarily differentiable; test functions ϕ(x, t) ∈ C1

0 (R × [0, ∞));

the eq. is multiplied by ϕ and integrated over space and time.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Conservative hyperbolic systems. Weak Solutions.

Let w : R × [0, ∞) → Ω =

• Ω ⊂ Rp be the unknown field, and

f : Ω → Rp smooth (flux-function). Conservative hyperbolic systems ∂tw + ∂xf (w) = 0, w(x, 0) = w0(x). Definition w is called a weak solution for the above conservative system if the following identity holds for any ϕ ∈ C1

0 (R × [0, ∞)) .

∞ ∞

−∞

[w · ∂tϕ + f (w) · ∂xϕ] dxdt = −

∞

−∞

w(x, 0) · ϕ(x, 0).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

• Nonconserv. hyperb. systems. (LeFloch) Weak Solutions.

A : Ω =

• Ω ⊂ Rp → Rp,

w : R × [0, ∞) → Ω. Nonconservative Hyperbolic Systems ∂tw + A(w)∂xw = 0, w(x, 0) = w0(x), the matrix-valued field A = A(w) is given and has, for each value w, p real eigenvalues and a basis of p eigenvectors. Definition A BV function w = w(x, t) is called a weak solution if:

• w satisfies the equations in a classical sense at the regular points;
• the following generalization of the Rankine-Hugoniot jump

relation holds along every curve of discontinuity of w.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

• Nonconserv. hyperb. systems. (LeFloch) Weak Solutions.

A : Ω =

• Ω ⊂ Rp → Rp,

w : R × [0, ∞) → Ω. Nonconservative Hyperbolic Systems ∂tw + A(w)∂xw = 0, w(x, 0) = w0(x), the matrix-valued field A = A(w) is given and has, for each value w, p real eigenvalues and a basis of p eigenvectors. Definition A BV function w = w(x, t) is called a weak solution if:

• w satisfies the equations in a classical sense at the regular points;
• the following generalization of the Rankine-Hugoniot jump

relation holds along every curve of discontinuity of w.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Generalized Rankine-Hugoniot jump relation

Preparation Denote by w a point in the space Rp and let Φ = Φ(η, w1, w2) be a family of paths connecting two points w1, w2 such that: (a) Φ : [0, 1] × Rp × Rp → Rp; (b) Φ(0, w1, w2) = w1, Φ(1, w1, w2) = w2, ∀w1, w2 ∈ Rp; (c) ∃λ > 0 : |∂ηΦ(η, w1, w2)| ≤ λ |w1 − w2| , ∀w1, w2 ∈ Rp, ∀η; (d) ∃λ > 0, |∂ηΦ(η, w1, w2) − ∂ηΦ(η, v1, v2)| ≤ λ(|w1 − v1| + |w2 − v2|), ∀wi, vi ∈ Rp, i = 1, 2, ∀η ∈ [0, 1]. Example: Φ(η, w1, w2)) = ηw1 + (1 − η)w2.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Generalized Rankine-Hugoniot jump relation

The Rankine-Hugoniot generalized jump relation Any shock wave connecting two states wL, wR at the speed s = s(wL, wR), w(x, t) =

• wL,

x < st wR, x > st , must satisfy −s (wR − wL) +

1

A(Φ(σ, wL, wR))∂σΦ(σ, wL, wR)dσ = 0. Note: In the conservative case when A(w) = Df (w) for some flux-function f , this relation reduces to −s (wR − wL) + f (wR) − f (wL) = 0, which is independent of the paths Φ and is nothing but the standard jump relation.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

1.1. Problem Formulation 1.2. Weak Solutions for 1-D Hyperbolic Systems

Generalized Rankine-Hugoniot jump relation

The Rankine-Hugoniot generalized jump relation Any shock wave connecting two states wL, wR at the speed s = s(wL, wR), w(x, t) =

• wL,

x < st wR, x > st , must satisfy −s (wR − wL) +

1

A(Φ(σ, wL, wR))∂σΦ(σ, wL, wR)dσ = 0. Note: In the conservative case when A(w) = Df (w) for some flux-function f , this relation reduces to −s (wR − wL) + f (wR) − f (wL) = 0, which is independent of the paths Φ and is nothing but the standard jump relation.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Recall our problem: nonconservativity and hyperbolicity.

Our Riemann Problem ∂t(θh) + ∂x(θhu) = 0, ∂t(θhu) + ∂x(θhu2) + θhg∂x(h) = 0, ∂t(θ) = 0, with initial condition (θ, h, u)|t=0 =

• (θL, hL, uL),

x < 0, (θR, hR, uR), x > 0. Eigenvalues-eigenvectors of the Jacobian matrix: λ1 = u − √gh, λ2 = u + √gh, λ3 = 0, r1 =

  

h −√gh

   ,

r2 =

  

h √gh

   ,

r3 =

  

−hu2 ghu (u2 − gh)θ

   .

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions

Let w := (h, u, θ)T, w(x, t) = wL + H(x − st)(wR − wL) a weak solution of our problem. For Φ(η, wL, wR) = wL + η(wR − wL), the generalized R-H jump relations give −s [|θh|] + [|θhu|] = 0, −s [|θhu|] +

• θh
• u2 + g h

2

• − g

2 [|θ|]

• hhL + [|h|]2

3

• = 0,

−s [|θ|] = 0. Possibilities: Case 1. [|θ|] = 0: constant porosity; Case 2. s = 0: stationary discontinuity.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions

Let w := (h, u, θ)T, w(x, t) = wL + H(x − st)(wR − wL) a weak solution of our problem. For Φ(η, wL, wR) = wL + η(wR − wL), the generalized R-H jump relations give −s [|θh|] + [|θhu|] = 0, −s [|θhu|] +

• θh
• u2 + g h

2

• − g

2 [|θ|]

• hhL + [|h|]2

3

• = 0,

−s [|θ|] = 0. Possibilities: Case 1. [|θ|] = 0: constant porosity; Case 2. s = 0: stationary discontinuity.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 1: constant porosity.

−s [|h|] + [|hu|] = 0, −s [|hu|] +

• h
• u2 + g h

2

• = 0.
• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 1: constant porosity.

−s [|h|] + [|hu|] = 0, −s [|hu|] +

• h
• u2 + g h

2

• = 0.

Hugoniot curves: H±

            

u − u0 = ±(h − h0)

• g

2

1

h + 1 h0

• ,

s = u0 ±

• g

2

• h + h2

h0

• .
• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 1: constant porosity.

−s [|h|] + [|hu|] = 0, −s [|hu|] +

• h
• u2 + g h

2

• = 0.

Lax condition selects the admissible branch of the Hugoniot curves S1(w, w0)

            

u − u0 = −(h − h0)

• g

2

1

h + 1 h0

• ,

h > h0, s = u0 −

• g

2

• h + h2

h0

• ,

λ1(u, h) < s < λ1(u0, h0), S2(w, w0)

            

u − u0 = (h − h0)

• g

2

1

h + 1 h0

• ,

h < h0, s = u0 +

• g

2

• h + h2

h0

• ,

λ2(u, h) < s < λ2(u0, h0).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 2: stationary discontinuity.

[|θhu|] = 0,

• θh
• u2 + g h

2

• − g

2 [|θ|]

• hh0 + [|h|]2

3

• = 0.
• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 2: stationary discontinuity.

[|θhu|] = 0,

• θh
• u2 + g h

2

• − g

2 [|θ|]

• hh0 + [|h|]2

3

• = 0.

Adimensionalize u, h, θ to get the solution u = 1 hθ, and 6Fr2 = hθ(6Fr2 − (h − 1)(h(2θ + 1) + θ + 2)), with Fr2 = u2

0/gh0.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 2: stationary discontinuity.

6Fr2 = hθ(6Fr2 − (h − 1)(h(2θ + 1) + θ + 2))

• Ψ(h,θ,Fr)

.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Discontinuous solutions. Case 2: stationary discontinuity.

6Fr2 = hθ(6Fr2 − (h − 1)(h(2θ + 1) + θ + 2))

• Ψ(h,θ,Fr)

. Proposition: Existence of the solution Let hm(θ, Fr) is the positive root of Ψh. Then: a) hm ≤ 1, if 2Fr2 ≤ θ + 1 and hm > 1, if 2Fr2 > θ + 1; b) if θ < 1, Fr = 1, and Ψ(hm(θ, Fr), θ, Fr) > 6Fr2, then there exist two solutions h1(θ, Fr), h2(θ, Fr) and

h1(θ, Fr) < hm(θ, Fr) < h2(θ, Fr) < 1, if 2Fr2 < θ + 1; 1 < h1(θ, Fr) < hm(θ, Fr) < h2(θ, Fr), if 2Fr2 > θ + 1;

c) if θ > 1 then there exist two solutions h1(θ, Fr), h2(θ, Fr) and

h1(θ, Fr) < hm(θ, Fr) < 1 < h2(θ, Fr), if 2Fr2 < θ + 1; h1(θ, Fr) < 1 < hm(θ, Fr) < h2(θ, Fr), if 2Fr2 > θ + 1.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Continuous solutions

Seeking for continuous solutions w(x, t) that depend ξ = x/t: (−ξI + A) dw dξ = 0.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Continuous solutions

Seeking for continuous solutions w(x, t) that depend ξ = x/t: (−ξI + A) dw dξ = 0.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Continuous solutions

Seeking for continuous solutions w(x, t) that depend ξ = x/t: (−ξI + A) dw dξ = 0. Solution: ξ = λi(w(ξ)) and dw dξ = ri(w(ξ)) ∇λi · ri(w(ξ)).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

2.1. Shock Waves 2.2. Rarefaction Waves

Continuous solutions

Seeking for continuous solutions w(x, t) that depend ξ = x/t: (−ξI + A) dw dξ = 0. R1(w, w0)

  

u − u0 = −2√g( √ h − √h0), h < h0, u = u0 + 2 3

x

t − λ1(u0, h0)

• , λ1(u0, h0) ≤ x

t ≤ λ1(u, h), R2(w, w0)

  

u − u0 = 2√g( √ h − √h0), h > h0, u = u0 + 2 3

x

t − λ2(u0, h0)

• , λ2(u0, h0) ≤ x

t ≤ λ2(u, h).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Riemann problem: the solutions for constant porosity.

Figure: Solution branches when θL = θR.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Result: Dam Break Problem.

Theorem: Existence of the solution Let u|t=0 = 0, h|t=0 =

• hL,

x < 0 hR, x > 0 , θ =

• θL,

x < 0 θR, x > 0 . Assuming that hL > hR, there exists a solution for the Riemann problem of the following structure:

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Result: Dam Break Problem.

Theorem: Existence of the solution Let u|t=0 = 0, h|t=0 =

• hL,

x < 0 hR, x > 0 , θ =

• θL,

x < 0 θR, x > 0 . Assuming that hL > hR, there exists a solution for the Riemann problem of the following structure: (i) when θL < θR, then ∃η1(θR/θL), η2(θR/θL) s.t. S2(UR, U2

I ) ⊕ R1(

U2

I , U1 I ) ⊕ J (U1 I ,

UL) ⊕ R1( UL, UL), if hR < η2, S2(UR, U2

I ) ⊕ S1(U2 I , U1 I ) ⊕ J (U1 I ,

UL) ⊕ R1( UL, UL), if η2 < hR < η1,

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Result: Dam Break Problem.

Theorem: Existence of the solution Let u|t=0 = 0, h|t=0 =

• hL,

x < 0 hR, x > 0 , θ =

• θL,

x < 0 θR, x > 0 . Assuming that hL > hR, there exists a solution for the Riemann problem of the following structure: (i) when θL < θR, then ∃η1(θR/θL), η2(θR/θL) s.t. S2(UR, U2

I ) ⊕ R1(

U2

I , U1 I ) ⊕ J (U1 I ,

UL) ⊕ R1( UL, UL), if hR < η2, S2(UR, U2

I ) ⊕ S1(U2 I , U1 I ) ⊕ J (U1 I ,

UL) ⊕ R1( UL, UL), if η2 < hR < η1, (ii) when θL > θR, S2(UR, U∗) ⊕ J (U∗, U∗) ⊕ R1(U∗, UL).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Result: Dam Break Problem.

Theorem: Existence of the solution Let u|t=0 = 0, h|t=0 =

• hL,

x < 0 hR, x > 0 , θ =

• θL,

x < 0 θR, x > 0 . Assuming that hL > hR, there exists a solution for the Riemann problem of the following structure: (i) when θL < θR, then ∃η1(θR/θL), η2(θR/θL) s.t. S2(UR, U2

I ) ⊕ R1(

U2

I , U1 I ) ⊕ J (U1 I ,

UL) ⊕ R1( UL, UL), if hR < η2, S2(UR, U2

I ) ⊕ S1(U2 I , U1 I ) ⊕ J (U1 I ,

UL) ⊕ R1( UL, UL), if η2 < hR < η1, (ii) when θL > θR, S2(UR, U∗) ⊕ J (U∗, U∗) ⊕ R1(U∗, UL).

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Riemann problem: the solutions for variable porosity.

hL I U U U

R

η h h

R

η

1 2 L I 2 I 1 2

UI h u

Figure: Solution branches when θL < θR.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

3.1. Solutions for Riemann Problem with Constant Porosity 3.2. Solutions for Dam Break Problem with Variable Porosity

Riemann problem: the solutions for variable porosity.

θ θ hL Fr=1 h

L R R

θ u h Fr=1 Figure: Solution branches when θL > θR.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

Riemann problem: variable porosity.

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

analytic solution numerical solution 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

analytic solution numerical solution

Figure: Dam break: θL < θR and hL > hR

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

Riemann problem: the influence of porosity.

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

variable porosity constant porosity 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

variable porosity constant porosity

Figure: Dam break: θL ≤ θR and hL > hR

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

Riemann problem: the influence of porosity.

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

Figure: Dam break: θL < θR (left fig) and θL > θR (right fig.)

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

The influence of vegetation on the water flow.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

The influence of vegetation on the water flow. Mathematical interest:

the model w. porosity - an exm of hyperb system w. var coeffs; dependence on space for the coefficients of the system

→ difficulties in defining weak solution; → new behavior into the structure of the sol.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

The influence of vegetation on the water flow. Mathematical interest:

the model w. porosity - an exm of hyperb system w. var coeffs; dependence on space for the coefficients of the system

→ difficulties in defining weak solution; → new behavior into the structure of the sol.

Practical interest:

software based on numerical methods to solve Riemann Probl

→ what do I do when there are no solutions? → can I always use these methods?

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

The influence of vegetation on the water flow. Mathematical interest:

the model w. porosity - an exm of hyperb system w. var coeffs; dependence on space for the coefficients of the system

→ difficulties in defining weak solution; → new behavior into the structure of the sol.

Practical interest:

software based on numerical methods to solve Riemann Probl

→ what do I do when there are no solutions? → can I always use these methods?

We have built a software to solve the 2-D shallow water model with porosity, topography and friction

it is not based on solving a Riemann Problem, but on a numerical scheme;

• ur algorithm works well for case (i), but not for (ii).
• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

The influence of vegetation on the water flow. Mathematical interest:

the model w. porosity - an exm of hyperb system w. var coeffs; dependence on space for the coefficients of the system

→ difficulties in defining weak solution; → new behavior into the structure of the sol.

Practical interest:

software based on numerical methods to solve Riemann Probl

→ what do I do when there are no solutions? → can I always use these methods?

We have built a software to solve the 2-D shallow water model with porosity, topography and friction

it is not based on solving a Riemann Problem, but on a numerical scheme;

• ur algorithm works well for case (i), but not for (ii).
• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

Our software on Riemann Problem with variable porosity

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

analytic solution numerical solution 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

analytic solution numerical solution

Figure: Dam break: θL < θR (left fig.) and θL > θR (right fig.)

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

Our software on Riemann Problem with constant porosity

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

analytic solution numerical solution 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

h x

analytic solution numerical solution

Figure: Dam break: θL = θR and hL > hR

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

Thank you for your kind attention!

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

References

• S. Ion, D. Marinescu, S.G. Cruceanu, V. Iordache, A data porting

tool for coupling models with different discretization needs, Envirnomental Modelling & Software, 62(2014), pp. 240–252, DOI: 10.1016/j.envsoft.2014.09.012.

• S. Ion, D. Marinescu, S.G. Cruceanu, Mathematical modeling of

the erosion processes in hydrographic basins, 9th International Symposium on Advanced Topics in Electrical Engineering, ATEE 2015, pp. 552–555, DOI: 10.1109/ATEE.2015.7133868. M.J. Baptist, V. Babovic, J. Rodriguez, Uthurburu, M. Keijzer, R.E. Uittenbogaard, A. Mynett and A. Verwey, On inducing equations for vegetation resistance, Journal of Hydraulic Research, 45:4(2007), pp. 435–450.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity

• 1. Introduction
• 2. Shock and Rarefaction Waves
• 3. Dam Break Problem
• 4. Final Remarks

4.1. Numerical Examples 4.2. Conclusions

References

Christophe Berthon, Frédéric Coquel and Philippe G. LeFloch, Why many theories of shock waves are necessary: kinetic relations for non-conservative systems. Proc. of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), pp 1-37.

• S. Ion, D. Marinescu, A.V. Ion, S.G. Cruceanu, V. Iordache, Water

flow on vegetated hill. 1D shallow water equation type model, Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica 07/2015; 23(3), pp. 83–96. DOI:10.1515/auom-2015-0049.

• S. Ion, D. Marinescu, S.G. Cruceanu

Riemann Problem for Shallow Water Equations with Porosity