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 36 th “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 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks The System of Equations and Riemann Problem 1-D Shallow Water Equations ∂ t ( θ h ) + ∂ x ( θ hu ) = 0 , ∂ t ( θ hu ) + ∂ x ( θ hu 2 ) + θ hg ∂ 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 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks The System of Equations and Riemann Problem Riemann Problem Solve ∂ t ( θ h ) + ∂ x ( θ hu ) = 0, ∂ t ( θ hu ) + ∂ x ( θ hu 2 ) + θ hg ∂ x ( h ) = 0, ∂ t ( θ ) = 0, with initial condition � ( θ L , h L , u L ) , x < 0, ( θ, h , u ) = ( θ R , h R , u R ) , x > 0. S. Ion, D. Marinescu, S.G. Cruceanu Riemann Problem for Shallow Water Equations with Porosity
1. Introduction 2. Shock and Rarefaction Waves 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks The System of Equations and Riemann Problem Riemann Problem Solve ∂ t ( θ h ) + ∂ x ( θ hu ) = 0, ∂ t ( θ hu ) + ∂ x ( θ hu 2 ) + θ hg ∂ x ( h ) = 0, ∂ t ( θ ) = 0, with initial condition � ( θ L , h L , u L ) , x < 0, ( θ, h , u ) = ( θ R , h R , u R ) , x > 0. S. Ion, D. Marinescu, S.G. Cruceanu Riemann Problem for Shallow Water Equations with Porosity
1. Introduction 2. Shock and Rarefaction Waves 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Conservative hyperbolic systems. Weak Solutions. ◦ Ω ⊂ R p be the unknown field , and Let w : R × [0 , ∞ ) → Ω = f : Ω → R p smooth ( flux-function ). Conservative hyperbolic systems ∂ t w + ∂ x f ( w ) = 0 , w ( x , 0) = w 0 ( x ) . S. Ion, D. Marinescu, S.G. Cruceanu Riemann Problem for Shallow Water Equations with Porosity
1. Introduction 2. Shock and Rarefaction Waves 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Conservative hyperbolic systems. Weak Solutions. ◦ Ω ⊂ R p be the unknown field , and Let w : R × [0 , ∞ ) → Ω = f : Ω → R p smooth ( flux-function ). Conservative hyperbolic systems ∂ t w + ∂ x f ( w ) = 0 , w ( x , 0) = w 0 ( x ) . Nonlinear hyperbolic problems: notion of classical solution; occurrence 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 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Conservative hyperbolic systems. Weak Solutions. ◦ Ω ⊂ R p be the unknown field , and Let w : R × [0 , ∞ ) → Ω = f : Ω → R p smooth ( flux-function ). Conservative hyperbolic systems ∂ t w + ∂ x f ( w ) = 0 , w ( x , 0) = w 0 ( x ) . Nonlinear hyperbolic problems: notion of classical solution; occurrence of discontinuities in the solution; need: new def. of a sol. w not necessarily differentiable; test functions ϕ ( x , t ) ∈ C 1 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 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Conservative hyperbolic systems. Weak Solutions. ◦ Ω ⊂ R p be the unknown field , and Let w : R × [0 , ∞ ) → Ω = f : Ω → R p smooth ( flux-function ). Conservative hyperbolic systems ∂ t w + ∂ x f ( w ) = 0 , w ( x , 0) = w 0 ( x ) . Definition w is called a weak solution for the above conservative system if the following identity holds for any ϕ ∈ C 1 0 ( R × [0 , ∞ )) . � ∞ � ∞ � ∞ [ w · ∂ t ϕ + f ( w ) · ∂ x ϕ ] dxdt = − w ( x , 0) · ϕ ( x , 0) . 0 −∞ −∞ S. Ion, D. Marinescu, S.G. Cruceanu Riemann Problem for Shallow Water Equations with Porosity
1. Introduction 2. Shock and Rarefaction Waves 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Nonconserv. hyperb. systems. (LeFloch) Weak Solutions. ◦ Ω ⊂ R p → R p , A : Ω = w : R × [0 , ∞ ) → Ω. Nonconservative Hyperbolic Systems ∂ t w + A ( w ) ∂ x w = 0 , w ( x , 0) = w 0 ( 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 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Nonconserv. hyperb. systems. (LeFloch) Weak Solutions. ◦ Ω ⊂ R p → R p , A : Ω = w : R × [0 , ∞ ) → Ω. Nonconservative Hyperbolic Systems ∂ t w + A ( w ) ∂ x w = 0 , w ( x , 0) = w 0 ( 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 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Generalized Rankine-Hugoniot jump relation Preparation Denote by w a point in the space R p and let Φ = Φ ( η, w 1 , w 2 ) be a family of paths connecting two points w 1 , w 2 such that: (a) Φ : [0 , 1] × R p × R p → R p ; ∀ w 1 , w 2 ∈ R p ; (b) Φ (0 , w 1 , w 2 ) = w 1 , Φ (1 , w 1 , w 2 ) = w 2 , (c) ∃ λ > 0 : | ∂ η Φ ( η, w 1 , w 2 ) | ≤ λ | w 1 − w 2 | , ∀ w 1 , w 2 ∈ R p , ∀ η ; (d) ∃ λ > 0 , | ∂ η Φ ( η, w 1 , w 2 ) − ∂ η Φ ( η, v 1 , v 2 ) | ≤ λ ( | w 1 − v 1 | + | w 2 − v 2 | ) , ∀ w i , v i ∈ R p , i = 1 , 2 , ∀ η ∈ [0 , 1]. Example: Φ ( η, w 1 , w 2 )) = η w 1 + (1 − η ) w 2 . S. Ion, D. Marinescu, S.G. Cruceanu Riemann Problem for Shallow Water Equations with Porosity
1. Introduction 2. Shock and Rarefaction Waves 1.1. Problem Formulation 3. Dam Break Problem 1.2. Weak Solutions for 1-D Hyperbolic Systems 4. Final Remarks Generalized Rankine-Hugoniot jump relation The Rankine-Hugoniot generalized jump relation Any shock wave connecting two states w L , w R at the speed � w L , x < st s = s ( w L , w R ), w ( x , t ) = , must satisfy w R , x > st � 1 − s ( w R − w L ) + A ( Φ ( σ, w L , w R )) ∂ σ Φ ( σ, w L , w R ) d σ = 0 . 0 Note: In the conservative case when A ( w ) = D f ( w ) for some flux-function f , this relation reduces to − s ( w R − w L ) + f ( w R ) − f ( w L ) = 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
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