Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems In memoriam of Professor Saul Abarbanel An Embedded Cartesian Scheme for the Navier-Stokes Equations Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering 18-20 December, 2018, Advances in Applied Mathematics, Tel Aviv University Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Joint work with M. Ben-Artzi, The Hebrew University J.-P . Croisille, University of Lorraine, France Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Outline 1. Navier-Stokes equations in streamfunction formulation 2. The one dimensional problem 3. Fourth order schemes in 2D regular domains 4. Fourth-order schemes for the N-S problem in irregular domains 5. Eigenvalues and Eigenfunctions of Biharmonic Problems Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Navier-Stokes Equations in Pure Streamfunction Formulation (Lagrange 1768) Let u ( x , t ) = ∇ ⊥ ψ , where ψ is the streamfunction. Then ∂ t (∆ ψ ) + ( ∇ ⊥ ψ ) · ∇ (∆ ψ ) = ν ∆ 2 ψ, in Ω . The boundary and initial conditions are ψ ( x, y, t ) = ∂ψ ∂n ( x, y, t ) = 0 , ( x, y ) ∈ ∂ Ω , ψ 0 ( x, y ) = ψ ( x, y, t ) | t =0 , ( x, y ) ∈ Ω . There is no need for vorticity boundary conditions. (*) Goodrich-Gustafson-Halasi, JCP (1990). [1] M. Ben-Artzi, J.-P . Croisille, D. Fishelov and S. Trachtenberg, J. Comp. Phys. 2005. Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Approximation in the one-dimensional case Consider the problem � ψ (4) ( x ) = f ( x ) , 0 < x < 1 (1) ψ (0) = 0 , ψ (1) = 0 , ψ ′ (0) = 0 , ψ ′ (1) = 0 . We lay out a uniform grid x 0 , x 1 , ..., x N where x i = ih and h = 1 /N . We approximate ψ on [ x i − 1 , x i +1 ] by a polynomial of degree 4, Q ( x ) = a 0 + a 1 ( x − x i ) + a 2 ( x − x i ) 2 + a 3 ( x − x i ) 3 + a 4 ( x − x i ) 4 , with interpolating values ψ i − 1 , ψ i , ψ i +1 , ψ x,i − 1 , ψ x,i +1 , where ψ x,i − 1 , ψ x,i +1 are approximate values for ψ ′ ( x i − 1 ) , ψ ′ ( x i +1 ) , which will be determined by the system as well. Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Approximation in the one-dimensional case We obtain ( a ) a 0 = ψ i , a 1 = 3 2 δ x ψ i − 1 ( b ) 4( ψ x,i +1 + ψ x,i − 1 ) , x ψ i − 1 a 2 = δ 2 ( c ) 2( δ x ψ x ) i , (2) � 1 � a 3 = 1 4( ψ x,i +1 + ψ x,i − 1 ) − 1 ( d ) 2 δ x ψ i h 2 � � 1 ( δ x ψ x ) i − δ 2 ( e ) a 4 = x ψ i . 2 h 2 Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Approximation in the one-dimensional case The approximate value ψ x,i is chosen as Q ′ ( x i ) . Thus, = a 1 = 3 2 δ x ψ i − 1 def ψ x,i 4( ψ x,i +1 + ψ x,i − 1 ) . This yields the Pad´ e approximation 1 6 ψ x,i − 1 + 2 3 ψ x,i + 1 6 ψ x,i +1 = δ x ψ i , 1 ≤ i ≤ N − 1 . (3) A natural approximation to ψ (4) ( x i ) is therefore Q (4) ( x i ) . Thus, � � = 24 a 4 = 12 def δ 4 ( δ x ψ x ) i − δ 2 x ψ i x ψ i . (4) h 2 Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Approximation in the one-dimensional case An approximation for the one-dimensional biharmonic problem is x ˜ δ 4 ( a ) ψ i = f ( x i ) 1 ≤ i ≤ N − 1 , σ x ˜ ψ x,i = δ x ˜ ( b ) ψ i , 1 ≤ i ≤ N − 1 , (5) ψ 0 = 0 , ˜ ˜ ψ N = 0 , ˜ ψ x, 0 = 0 , ˜ ( c ) ψ x,N = 0 . where σ x ϕ = 1 6 ϕ i − 1 + 2 3 ϕ i + 1 6 ϕ i +1 . Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Consistency of the three-point biharmonic operator Proposition Suppose that ψ ( x ) is a smooth function on [0 , 1] . Then, • � � δ 4 i − ( ψ (4) ) ∗ ( x i ) | ≤ Ch 4 � ψ (8) � L ∞ , 2 ≤ i ≤ N − 2 . x ψ ∗ | σ x (6) • At near boundary points i = 1 and i = N − 1 , the fourth order accuracy of (6) drops to first order, � � δ 4 x ψ ∗ i − ( ψ (4) ) ∗ ( x i ) | ≤ Ch � ψ (5) � L ∞ , i = 1 , N − 1 . | σ x (7) Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Optimal convergence of the three-point biharmonic operator The following error estimate holds. Theorem Let ˜ ψ be the approximate solution of the biharmonic problem and let ψ be the exact solution and ψ ∗ its evaluation at grid points. The error ψ − ψ ∗ = δ − 4 x f ∗ − ( ∂ − 4 x f ) ∗ satisfies e = ˜ 1 ≤ i ≤ N − 1 | e i | ≤ Ch 4 , | e | h ≤ Ch 4 , max (8) where C depends only on f . [2] M. Ben-Artzi, J.-P . Croisille and D. Fishelov, Navier-Stokes Eqns. in Planar Domains, 2013, Imperial College Press. J. Scientific Computing, 2012. B. Gustafsson,1981,S. Abarbanel, A. Ditkowski and B. Gustafsson,2000, M. Svard and J. Nordstrom,2006 Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems Linear time-independent equation- constant coefficients case Consider an invertible problem u (4) + au (2) + bu = f, x ∈ [0 , 1] , (9) (with boundary conditions on u, u ′ ) and its approximation δ 4 x v + a ˜ δ 2 x v + b v = f ∗ , (10) x v − δ x v x . Then, the error e = v − u ∗ satisfies where ˜ δ 2 x v = 2 a 2 = 2 δ 2 | e ( t ) | h ≤ Ch 4 , (11) where C > 0 depends only on f . [3] M. Ben-Artzi, J.-P . Croisille, D. Fishelov and R. Katzir, IMA J. Numer. Anal, 2017. Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system A high order scheme for irregular domains Eigenvalues and Eigenfunctions of Biharmonic Problems The linear evolution equation Consider ∂ t u = − ∂ 4 x u + a∂ 2 x u + bu, x ∈ [0 , 1] , t ≥ 0 . (12) with the initial condition u ( t = 0) = u 0 , and its approximation v t = − δ 4 x v + a ˜ δ 2 x v + b v , t ≥ 0 . (13) Then the error e = v − u ∗ satisfies | e ( t ) | h ≤ Ch 4 − ǫ , t ∈ [0 , T ] , h < h 0 , (14) where C > 0 depends only on u 0 , T, ǫ. [4] M. Ben-Artzi, J.-P . Croisille and D. Fishelov, submitted. Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering An Embedded Cartesian Scheme for the Navier-Stokes Equations
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