Very high-order finite volume scheme for the 2D linear - PowerPoint PPT Presentation

Very high-order finite volume scheme for the 2D linear convection-diffusion problem S. Clain 1 , 3 , G.J. Machado 1 , J.M. Nobrega 2 , R.M.S. Pereira 1 , A. Boularas 4 1 Mathematical Centre, University of Minho, Portugal 2 Institute for Polymers and Composites/I3N, University of Minho, Portugal 3 Mathematical Institute, Paul Sabatier University, France 4 Laplace centre, Paul Sabatier University, France Ofir, 2014 april-28th may-2nd

Outline ➓ 2D linear convection-diffusion with finite volume ➓ Discretization and residual formulation ➓ Polynomial reconstructions ➓ Scheme design ➓ Numerical tests

Model Find φ ✏ ♣ φ 1 , φ 2 q on the bounded open domain Ω such that ∇ . ♣ V 1 φ 1 ✁ k 1 ∇ φ 1 q ✏ f 1 , in Ω 1 , (1a) ∇ . ♣ V 2 φ 2 ✁ k 2 ∇ φ 2 q ✏ f 2 , in Ω 2 , (1b) k 1 ∇ φ 1 . n Γ ✏ k 2 ∇ φ 2 . n Γ , on Γ , (1c) φ ✏ φ D , (1d) on Γ D , φ 1 ✏ φ 2 on Γ or k 1 ∇ φ 1 . n Γ ✏ h ♣ φ 2 ✁ φ 1 q .

Discretization ➺ ➺ ♣ V . n φ ✁ k ∇ φ. n q ds ✁ f dx ✏ 0 . (divergence theorem) ❇ c i c i ✑ ✙ ➳ ➳ R ⑤ e ij ⑤ ζ r V ♣ q ij , r q . n ij φ ♣ q ij , r q ✁ k ♣ q ij , r q ∇ φ ♣ q ij , r q . n ij j P ν ♣ i q r ✏ 1 ✁⑤ c i ⑤ f i ✏ O ♣ h 2 R i q . (Gauss Points)

Residual formulation Based on the previous expression: the residual formulation is ➳ ➳ R ⑤ e ij ⑤ G i ✏ ζ r F ij , r ✁ f i , ⑤ c i ⑤ r ✏ 1 j P ν ♣ i q where F ij , r ✓ V ♣ q ij , r q . n ij φ ♣ q ij , r q ✁ k ♣ q ij , r q ∇ φ ♣ q ij , r q . n ij , approximation of the flux at the Gauss point q ij , r . ☞ Sixth-order approximation for F ij , r .

Polynomial reconstructions

Conservative reconstruction for cells c i : cell of mesh T h , d : the polynomial degree, S ♣ c i , d q : the associated stencil, φ i : mean value on c i . ✦ ✮ ➳ ♣ x ✁ b i q α ✁ M α ♣ R d ,α φ i ♣ x ; d q ✏ φ i � i i 1 ↕⑤ α ⑤↕ d α ✏ ♣ α 1 , α 2 q , ⑤ α ⑤ ✏ α 1 � α 2 , x ✏ ♣ x 1 , x 2 q , b i the centroid of cell c i ➺ i ✏ 1 ♣ x ✁ b i q α dx to provide the conservation. ☞ Set M α ⑤ c i ⑤ c i

Coefficients for ♣ φ i i vector gathering coefficients R d ,α R d i Assume mean values φ ℓ on cells c ℓ , ℓ P S ♣ c i , d q are known, ♣ R d i minimizes the functional ➺ ✑ 1 ✙ 2 ➳ ♣ E i ♣ R d i ; d q ✏ φ i ♣ x ; d q dx ✁ φ ℓ , ⑤ c ℓ ⑤ c ℓ ℓ P S ♣ c i , d q ☞ Lead to an over-determined linear system A d i R d i ✏ b S i where b S i represents the variations φ ℓ ✁ φ i , ℓ P S ♣ c i ; d q

Preconditioning and solving Determine the Moore-Penrose pseudo-inverse matrix for system ♣ A d i P d i q♣ P d i q ✁ 1 R d i ✏ b S i with the diagonal matrix P d i ✏ diag ♣⑤ c i ⑤ ✁⑤ α ⑤④ 2 q 1 ↕⑤ α ⑤↕ d . Motivation: the A d i matrix coefficients are ➺ ➺ 1 ♣ x ✁ b i q α dx ✁ 1 ♣ x ✁ b i q α dx . ⑤ c ℓ ⑤ ⑤ c i ⑤ c ℓ c i ☞ Strongly reduces the effect of the power α . i q ✿ and get We compute the pseudo inverse matrix ♣ A d i P d R d i ✏ P d i ♣ A d i P d i q ✿ b S i .

Conservative reconstruction for Γ e ij ⑨ Γ : edge on the interface, c j the cell on the Ω 2 side, d : the polynomial degree, S ♣ c j , d q : the associated stencil. ✦ ✮ ➳ ♣ x ✁ b j q α ✁ M α q R d ,α φ j ♣ x ; d q ✏ φ j � . j j 1 ↕⑤ α ⑤↕ d b j the centroid of cell c j . ➺ j ✏ 1 ♣ x ✁ b j q α dx to provide the conservation. ☞ Set M α ⑤ c j ⑤ c j ☞ Only use the cells on the ” j ” side.

Coefficients for q φ j j vector gathering coefficients R d ,α R d j Assume mean values φ ℓ on cells c ℓ , ℓ P S ♣ e ij , d q and φ ij the mean value on e ij are known, q R d j minimizes the functional ➺ ✑ 1 ✙ 2 ➳ q E j ♣ R d j ; d q ✏ φ j ♣ x ; d q dx ✁ φ ℓ ⑤ c ℓ ⑤ c ℓ ℓ P S ♣ c j , d q ➺ ✑ 1 ✙ 2 q � ω ij φ j ♣ x ; d q ds ✁ φ ij ⑤ e ij ⑤ e ij with ω ij a positive weight.

Conservative reconstruction for Γ D e iD : edge on the boundary Γ D , d : the polynomial degree, S ♣ e iD , d q : the associated stencil, ➺ 1 φ iD ✏ φ D ♣ s q ds . ⑤ e iD ⑤ e iD ✦ ✮ ➳ ♣ x ✁ m iD q α ✁ M α ♣ R d ,α φ iD ♣ x ; d q ✏ φ iD � , iD iD 1 ↕⑤ α ⑤↕ d m iD the centroid of edge e iD ➺ 1 ♣ x ✁ m iD q α ds to provide the conservation. ☞ Set M α iD ✏ ⑤ e iD ⑤ e iD

Coefficients for ♣ φ iD iD vector gathering coefficients R d ,α R d iD Assume mean values φ ℓ on cells c ℓ , ℓ P S ♣ e iD , d q are known, ♣ R d iD minimizes the functional ➺ ✑ 1 ✙ 2 ➳ ♣ E iD ♣ R d iD ; d q ✏ φ iD ♣ x ; d q dx ✁ φ ℓ ω iD ,ℓ , ⑤ c ℓ ⑤ c ℓ ℓ P S ♣ e iD , d q with ω iD ,ℓ positive weights. ☞ Coefficients ω iD ,ℓ are very important to provide ”good” properties.

Non conservative reconstruction for inner edges e ij : inner edge of the mesh, d : the polynomial degree, S ♣ e ij , d q : the associated stencil, no value associated to e ij . ➳ r R d ,α ij ♣ x ✁ m ij q α φ ij ♣ x ; d q ✏ 0 ↕⑤ α ⑤↕ d m ij the centroid of edge e ij ☞ Coefficient R d ,α for ⑤ α ⑤ ✏ 0 is also unknown. ij

Coefficients for r φ ij ij vector gathering coefficients R d ,α R d ij Assume mean values φ ℓ on cells c ℓ , ℓ P S ♣ e ij , d q are known, r R d ij minimizes the functional ➺ ✑ 1 ✙ 2 ➳ r E ij ♣ R d ij ; d q ✏ ω ij ,ℓ φ ij ♣ x ; d q dx ✁ φ ℓ , ⑤ c ℓ ⑤ c ℓ ℓ P S ♣ e ij , d q where ω ij ,ℓ are positive weights. ☞ One more time: coefficients ω ij ,ℓ are very important.

Polynomial Reconstruction Operators ✌ Φ ✏ ♣ φ i q i P C the mean values vector. ✌ Operators Φ Ñ ♣ φ i , q φ j , ♣ φ iD , and r φ ij are linear. ✌ φ polynomial function of degree d , φ i the exact mean values. d -exact reconstruction if φ i ♣ x ; d q ✏ q ♣ φ j ♣ x ; d q ✏ ♣ φ iD ♣ x ; d q ✏ r φ ij ♣ x ; d q ✏ φ ♣ x q , x P R 2 . ☞ The finite volume method associated to the polynomial reconstruction is a d � 1 th -order method.

The flux on edge (except Γ ) 1. e ij is an inner edge (not on Γ ), F ij , r ✏ r V ♣ q ij , r q . n ij s � ♣ φ i ♣ q ij , r ; d q � r V ♣ q ij , r q . n ij s ✁ ♣ φ j ♣ q ij , r ; d q ✁ k ♣ q ij , r q ∇ r φ ij ♣ q ij , r ; d q . n ij . 2. e iD belongs to Γ D , F iD , r ✏ r V ♣ q iD , r q . n iD s � ♣ φ i ♣ q iD , r ; d q � r V ♣ q iD , r q . n iD s ✁ φ D ♣ q iD , r q ✁ k ♣ q iD , r q ∇ ♣ φ iD ♣ q iD , r ; d q . n iD .

The flux on edge of e ij ⑨ Γ ✌ Transfer condition: F ij , r ✏ h ♣ q ij , r qr ♣ φ i ♣ q ij , r ; d q ✁ ♣ φ j ♣ q ij , r ; d qs . ✌ Continuity condition: we perform three steps Step 1: compute the reconstructions ♣ φ i ♣ x ; d q for all c i ⑨ Ω 1 . ➺ 1 ♣ Step 2: compute φ ij ✏ φ i ♣ x ; d q ds for all e ij ⑨ Γ . ⑤ e ij ⑤ e ij Step 3: F ij , r ✏ k 2 ♣ q ij , r q ∇ q φ j ♣ q ij , r ; d q . n ij .

Resolution ① The polynomial reconstruction operators are linear. ② The flux computations are linear. ③ The residual expression is linear: Φ Ñ G i ♣ Φ q . We get a linear operator Φ Ñ G ♣ Φ q ✏ ♣ G 1 ♣ Φ q , ..., G I ♣ Φ qq . Problem: Find Φ such that G ♣ Φ q ✏ 0 . ☛ Matrix-free problem: use GMRES method. Preconditioning is very very important: P preconditioning matrix substitute Φ Ñ G ♣ Φ q by Φ Ñ PG ♣ Φ q .

Preconditioning matrix ☞ G ♣ Φ q ✏ A Φ ✁ b but we do not have matrix A : ILU not possible. Diagonal preconditioning matrix P ✏ D ✁ 1 with P ✒ k ♣ b i q ✚ ➳ D P ♣ i , i q ✏ 1 ⑤ b i b j ⑤ � r V ♣ m ij q . n ij s � ⑤ e ij ⑤ . ⑤ c i ⑤ j P ν ♣ i q More sophisticated preconditioning matrix, A P ✏ D P for the diagonal coefficients and ✒ ✚ A P ♣ i , j q ✏ ⑤ e ij ⑤ ✁ k ♣ m ij q ⑤ b i b j ⑤ � r V ♣ m ij q . n ij s ✁ , j P ν ♣ i q . ⑤ c i ⑤

Incomplete inverse of A P Preconditioning matrix is supposed to be P ✏ A ✁ 1 P Substitute with the incomplete inverse A ✿ P with the same non-null entries of A P . Taking advantage of the structure of A P and A ✿ P provides explicit construction of A ✿ P P ♣ i , j q ✏ ✁ A P ♣ i , j q A ✿ P ♣ i , i q A ✿ A P ♣ j , j q , j P ν ♣ i q with 1 A ✿ P ♣ i , i q ✏ . ➳ A P ♣ i , j q A P ♣ j , i q A P ♣ i , i q ✁ A P ♣ j , j q j P ν ♣ i q

Curved boundary treatment ☞ Problem: we set the boundary condition on the edge while it is prescribed on the curve. ✌ q iD , r Gauss points on e iD , ✌ p iD , r Gauss points on the curve, ♣ ✌ φ iD is evaluated using φ D ♣ q iD , r q and not φ D ♣ p iD , r q . ➳ R ☞ Idea: modify the mean value φ iD ✏ ζ r φ D ♣ q iD , r q ds but φ iD r ✏ 1 is still associated to edge e iD .