High-order compact schemes for the Navier-Stokes Equations Dalia - - PowerPoint PPT Presentation

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

High-order compact schemes for the Navier-Stokes Equations

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering 20-25 August, 2018, ICERM, Providence

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

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 High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Outline

• 1. Navier-Stokes equations in streamfunction formulation
• 2. Optimal convergence of a compact fourth-order scheme in 1D
• 3. The time-dependent problem in 1D
• 4. Fourth order schemes in 2D regular domains
• 5. Fourth-order schemes for the N-S problem in irregular domains

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

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∆2ψ, in Ω. The boundary and initial conditions are ψ(x, y, t) = ∂ψ ∂n (x, y, t) = 0, (x, y) ∈ ∂Ω, ψ0(x, y) = ψ(x, y, t)|t=0, (x, y) ∈ Ω. In contrast to the vorticity formulation, here 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 High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Approximation in the one-dimensional case Consider the problem ψ(4)(x) = f(x), 0 < x < 1 ψ(0) = 0, ψ(1) = 0, ψ′(0) = 0, ψ′(1) = 0. (1) We lay out a uniform grid x0, x1, ..., xN where xi = ih and h = 1/N. We approximate ψ on [xi−1, xi+1] by a polynomial of degree 4, Q(x) = a0 + a1(x − xi) + a2(x − xi)2 + a3(x − xi)3 + a4(x − xi)4, with interpolating values ψi−1, ψi, ψi+1, ψx,i−1, ψx,i+1, where ψx,i−1, ψx,i+1 are approximate values for ψ′(xi−1), ψ′(xi+1), which will be determined by the system as well.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Approximation in the one-dimensional case We obtain                      (a) a0 = ψi, (b) a1 = 3 2δxψi − 1 4(ψx,i+1 + ψx,i−1), (c) a2 = δ2

xψi − 1

2(δxψx)i, (d) a3 = 1 h2 1 4(ψx,i+1 + ψx,i−1) − 1 2δxψi

• (e)

a4 = 1 2h2

• (δxψx)i − δ2

xψi

• .

(2)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Approximation in the one-dimensional case The approximate value ψx,i is chosen as Q′(xi). Thus, ψx,i

def

= a1 = 3 2δxψi − 1 4(ψx,i+1 + ψx,i−1). This yields the Pad´ e approximation 1 6ψx,i−1 + 2 3ψx,i + 1 6ψx,i+1 = δxψi, 1 ≤ i ≤ N − 1. (3) A natural approximation to ψ(4)(xi) is therefore Q(4)(xi). Thus, δ4

xψi def

= 24a4 = 12 h2

• (δxψx)i − δ2

xψi

• .

(4)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Approximation in the one-dimensional case An approximation for the one-dimensional biharmonic problem is              (a) δ4

x ˜

ψi = f(xi) 1 ≤ i ≤ N − 1, (b) σx ˜ ψx,i = δx ˜ ψi, 1 ≤ i ≤ N − 1, (c) ˜ ψ0 = 0, ˜ ψN = 0, ˜ ψx,0 = 0, ˜ ψx,N = 0. (5) where σxϕ = 1 6ϕi−1 + 2 3ϕi + 1 6ϕi+1.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Consistency of the three-point biharmonic operator

Proposition

Suppose that ψ(x) is a smooth function on [0, 1]. Then,

• |σx
• δ4

xψ∗ i − (ψ(4))∗(xi)

• | ≤ Ch4ψ(8)L∞, 2 ≤ i ≤ N − 2.

(6)

• At near boundary points i = 1 and i = N − 1, the fourth order

accuracy of (6) drops to first order, |σx

• δ4

xψ∗ i − (ψ(4))∗(xi)

• | ≤ Chψ(5)L∞, i = 1, N − 1.

(7)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

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 e = ˜ ψ − ψ∗ = δ−4

x f ∗ − (∂−4 x f)∗ satisfies

max

1≤i≤N−1 |ei| ≤ Ch4,

|e|h ≤ Ch4, (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 High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Linear time-independent equation Consider an invertible problem u(4) + A(x)u(2) + A′(x)u′ + B(x)u = f, x ∈ [0, 1], (9) (with boundary conditions on u, u′) and its approximation δ4

xvi +A(xi)˜

δ2

xvi +A′(xi)vx,i +B(xi)vi = f(xi),

1 ≤ i ≤ N −1, (10) where ˜ δ2

xv = 2a2 = 2δ2 xv − δxvx. Then, vhk converges to u in C([0, 1]),

where vhk is the continuous piecewise linear function corresponding to v. [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 High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Linear time-independent equation- constant coefficients case Consider an invertible problem u(4) + au(2) + bu = f, x ∈ [0, 1], (11) (with boundary conditions on u, u′) and its approximation δ4

xv + a˜

δ2

xv + bv = f ∗,

(12) where ˜ δ2

xv = 2a2 = 2δ2 xv − δxvx. Then, the error e = v − u∗ satisfies

|e(t)|h ≤ Ch4, (13) 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 High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The linear evolution equation Consider ∂tu = −∂4

xu + a∂2 xu + bu,

x ∈ [0, 1], t ≥ 0. (14) with the initial condition u(t = 0) = u0, and its approximation vt = −δ4

xv + a˜

δ2

xv + bv,

t ≥ 0. (15) Then the error e = v − u∗ satisfies |e(t)|h ≤ Ch4−ǫ, t ∈ [0, T], h < h0, (16) where C > 0 depends only on u0, T, ǫ. [4] M. Ben-Artzi, J.-P . Croisille and D. Fishelov, submitted.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The linear evolution equation-sketch of the proof Consider ∂tu = −∂4

xu,

x ∈ [0, 1], t ≥ 0. (17) Applying ∂−4

x

• n the last equation,

∂t∂−4

x u = −u.

(18) By the optimal error bound for ∂−4

x ∂tu = −u we have

∂tδ−4

x u∗ = −u∗ + O(h4).

(19) Consider the approximation ∂tv = −δ4

xv and applying δ−4 x

• n the last

equation, we have ∂tδ−4

x v = −v.

(20) Then the error e = v − u∗ satisfies ∂tδ−4

x e(t) = −e(t) + O(h4).

(21)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The linear evolution equation-sketch of the proof Defining w = δ−4

x e

∂tw(t) = −e(t) + O(h4). (22) Inner multiplication with w(t) yields 1 2∂t|w(t)|2

h + (e(t), w(t))h = (O(h4), w(t))h.

(23) By the coercivity (e(t), w(t))h = (δ4

xw, w)h ≥ C|w(t)|2 h

∂t|w(t)|2

h + C|w(t)|2 h ≤ O(h8) + |w(t)|2 h.

(24) By Grownwall’s inequality |w(t)|h ≤ Ch4.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The linear evolution equation-sketch of the proof Going back to ∂tw(t) = −e(t) + O(h4). (25) Approximating ∂tw(t) by a finite difference scheme SQw, for which SQw(t) − w′(t) = O((∆t)Q), and choosing ∆t = h4/Q = hǫ, |e(t)|h ≤ Ch4−ǫ, t ∈ [0, T], h < h0, (26) where C > 0 depends only on u0, T, ǫ.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical results for time-dependent problems in 1D We consider the equation    ∂tu = −∂4

xu + a∂2 xu − bu + f(x, t),

0 < x < 1, t ≥ 0 u(0, t) = ∂xu(0, t) = 0, u(1, t) = ∂xu(1, t) = 0, t ≥ 0 u(x, 0) = u0(x), 0 ≤ x ≤ 1. (27) The semi-discrete analog to (27) is             

dvj(t) dt

= −δ4

xvj(t) + a ˜

δ2

xvj(t) − b vj(t) + f ∗ j (t),

j = 1, ..., N − 1, t ≥ 0, v0(t) = 0, vN(t) = 0, vx,0(t) = 0, vx,N(t) = 0, t ≥ 0 vj(0) = (u∗

0)j := u0(xj),

j = 0, ..., N. (28)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The exact solution is uε(x, t) = p(x) sin

• 1/qε(x)
• sin(2πt).

(29) The polynomial functions p(x) and qε(x) are given by p(x) = 16x2(1 − x)2, qε(x) = (x − 1/2)2 + ε, ε > 0. (30) In this example we took ε = 0.05. The results, even for the very coarse grid with N = 32, show the remarkable accuracy of the scheme. mesh N = 32 Rate N = 64 Rate N = 128 Rate N = 256 |e|∞ 1.23(-1) 6.70 1.18(-3) 4.27 6.11(-5) 4.06 3.65(-6)

Table: Equation (27) with a = 1, b = −1 with exact solution: u(x, t) = p(x) sin(1/qε(x)) sin(2πt), ε = 0.05. The initial time is t0 = 0 and the final time is tf = 0.75. The time step is ∆t = Kh2 with K = 384/5 (10 iterations for N = 32).

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical results for time-dependent problems in 1D

0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1 1.5

x v(x, tf)

0.2 0.4 0.6 0.8 1

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

x v(x, tf)

Figure: Exact and calculated solution u(x, t) = p(x) sin(1/qε(x)) sin(2πt) for Equation (27) ε = 0.05, at final time tf = 0.75. The time step is ∆t = 384

5 h2.

The number N of grid points was consecutively N = 32 and N = 256.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical results for time-dependent problems in 1D-Kuramuto-Sivashinsky Eqn. Consider the Kuramoto–Sivashinsky equation ∂tu = −∂4

xu − ∂2 xu − u∂xu + f,

0 < x < 1, t > 0, u(0, t) = ∂xu(0, t) = 0 = u(1, t) = ∂xu(1, t) = 0. (31) u(x, t) = e−tx4(1 − x)4, 0 < x < 1, t > 0. (32) Mesh N = 16 Rate N = 32 Rate N = 64 |e|h 4.6595(-4) 3.97 2.9750(-5) 3.99 1.8702(-6) |ex|h 2.2606(-3) 4.65 8.9829(-5) 4.24 4.7682(-6)

Table: Compact scheme for KS equation (31) with exact solution u = e−tx4(x − 1)4 on [0, 1]. We display |e|h and |ex|h, the errors in u, and ux, respectively at t = 0.25. The time step is ∆t = h2.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical results for time-dependent problems in 1D

Figure: Second KS numerical example: Exact solution (solid line) and computed solution (circles) for N = 32 (left) and N = 64 (center) .

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The biharmonic equation We provide second and fourth-order approximations to the biharmonic equation. ∆2ψ = f(x, y), (x, y) ∈ Ω, ψ = 0,

∂ψ ∂n = 0,

(x, y) ∈ ∂Ω. (33) We construct an interpolating polynomial that yields a fourth order compact scheme for the biharmonic operator.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The 23-point biharmonic operator Suppose the differential problem is given for x, y in Ω = [0, 1] × [0, 1] and lay out a uniform grid (xi, yj), 0 ≤ i, j ≤ N. Denoting by ˜ x = x − xi, ˜ y = y − yj, we approximate ψ(x, y) on [xi−1, xi+1] × [yj−1, yj+1] by P(x, y) = a0 + a1˜ x + a2˜ y + a3˜ x2 + a4˜ x˜ y + a5˜ y2 + a6˜ x3 + a7˜ x2˜ y + a8˜ x˜ y2 + a9˜ y3 + a10˜ x4 + a11˜ x3˜ y + a12˜ x2˜ y2 + a13˜ x˜ y3 + a14˜ y4 + a15˜ x5 + a16˜ x4˜ y + a17˜ x3˜ y2 + a18˜ x2˜ y3 + a19˜ x˜ y4 + a20˜ y5 + a21˜ x4˜ y2 + a22˜ x2˜ y4. (34)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Modified Stephenson’s Scheme

❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣

ψi−1,j−1, ψx,i−1,j−1 ψy,i−1,j−1 ψi,j−1 (ψy)i,j−1 ψi+1,j−1, ψx,i+1,j−1 ψy,i+1,j−1 ψi−1,j, (ψx)i−1,j ψi+1,j, (ψx)i+1,j ψi−1,j+1, ψx,i−1,j+1 ψy,i−1,j+1 ψi,j+1 (ψy)i,j+1 ψi+1,j+1, ψx,i+1,j+1 ψy,i+1,j+1 ψi,j ψx,i,j ψy,i,j

J.W. Stephenson, ”Single cell discretizations of order two and four for biharmonic problems”, J. Comp. Phys. 1984.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Fourth Order Spatial Discretization for the biharmonic operator δ4

xψi,j = 12

h2

• (δxψx)i,j − δ2

xψi,j

• ,

1 ≤ i, j ≤ N − 1. δ4

yψi,j = 12

h2

• (δyψy)i,j − δ2

yψi,j

• ,

1 ≤ i, j ≤ N − 1. The mixed term ψxxyy is approximated by ˜ δ2

xyψi,j = 3δ2 xδ2 yψi,j − δ2 xδyψy,i,j − δ2 yδxψx,i,j = ∂2 x∂2 yψi,j + O(h4)

[5] M. Ben-Artzi, J.-P . Croisille and D. Fishelov, SISC 2008.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The Navier-Stokes Eqs’: 4th-order Spatial Discretization The Laplacian of ψ: The Laplacian of ψ is approximated by ˜ ∆hψ, where ˜ ∆hψ = 2δ2

xψ − δxψx + 2δ2 yψ − δyψy,

δ2

xψi,j = ψi+1,j − 2ψi,j + ψi−1,j

h2 , δ2

yψi,j = ψi,j+1 − 2ψi,j + ψi,j−1

h2 and δxψi,j = ψi+1,j−ψi−1,j

2h

, δyψi,j = ψi,j+1−ψi,j−1

2h

.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The Convective Term The convective term −ψy∆xψ + ψx∆yψ is therefore approximated by ˜ Ch(ψ) = − ˜ ψy

• ∆h ˜

ψx + 5

2

• 6 δxψ− ˜

ψx h2

− δ2

x ˜

ψx

• + δxδ2

yψ − δxδy ˜

ψy

• + ˜

ψx

• ∆h ˜

ψy + 5

2

• 6 δyψ− ˜

ψy h2

− δ2

y ˜

ψy

• + δyδ2

xψ − δyδx ˜

ψx

• ,

where ˜ ψx and ˜ ψy are the 6-th order accurate Pad´ e approximations to ∂xψ and ∂yψ. [6] M. Ben-Artzi, J.-P . Croisille, D. Fishelov, Navier-Stokes Equations in Planar Domains, Imperial College Press, 2013. J. Scientific Computing, 2009.

• T. Hou and B. Wetton, 2009.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Fourth-order spatial discretization and an IMEX scheme

( ˜ ∆hψi,j)n+1/2−( ˜ ∆hψi,j)n ∆t/2

= − ˜ Ch(ψn)i,j + ν

2[ ˜

∆2

hψn+1/2 i,j

+ ˜ ∆2

hψn i,j] ( ˜ ∆hψi,j)n+1−( ˜ ∆hψi,j)n ∆t

= − ˜ Ch(ψn+1/2)i,j + ν

2[ ˜

∆2

hψn+1 i,j

+ ˜ ∆2

hψn i,j].

Note here that only the discrete Laplacian and biharmonic operators, which are approximated by a compact scheme, have to be inverted at each step.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Convergence of the semi-discrete scheme Theorem: Let ˜ ψ be the solution of ∂t∆h ψ = −∇⊥

h

ψ · (∆h∇h ψ) + ν∆2

h

ψ, and ψ is the exact solution of NS equations: ∂t∆ψ = −∇⊥ψ · ∇(∆ψ) + ν∆2ψ. Define the error e(t) as e(t) = ψ − ψ. Let T > 0. Then there exist constants C, h0 > 0, depending possibly on T, ν and the exact solution ψ, such that, for all 0 ≤ t ≤ T, |δ+

x e|2 h + |δ+ y e|2 h ≤ Ch3

, 0 < h ≤ h0. [7] M. Ben-Artzi, J.-P . Croisille, D. Fishelov, ”Convergence of a compact scheme for the pure streamfunction formulation of Navier-Stokes equations”, SINUM, 2006

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical Results in Two Dimensions Test problem ∂(∆ψ) ∂t + (∇⊥ψ) · ∇(∆ψ) = ν∆2ψ + f(x, y, t), for −1 ≤ x, y ≤ 1 with ν = 1, with exact solution ψ = e−t(1 − x2)3(1 − y2)3 Table 1 summarizes the error, e, the relative error, er, and the error in ψx in the l2 norm, where el2 = ψcomp − ψexactl2, er = el2/ψexactl2 (ex)l2 = (ψx)comp − (ψx)exactl2.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

mesh 17 × 17 Rate 33 × 33 Rate 65 × 65 t = 0.25 e 3.0525(-4) 4.02 1.8835(-5) 4.00 1.1734(-6) er 5.7441(-4) 3.5460(-5) 2.2092(-6) ex 1.7837(-4) 3.93 1.1670(-5) 3.98 7.3752(-7) t = 0.5 e 2.0085(-4) 4.00 1.2541(-5) 4.00 7.8361(-7) er 4.8536(-4) 3.0317(-5) 1.8944(-6) ex 1.9896(-4) 4.00 1.2436(-5) 4.00 7.7745(-7) t = 0.75 e 1.5508(-4) 4.00 9.6887(-6) 4.00 6.0551(-7) er 4.8119(-4) 3.0075(-5) 1.8796(-6) ex 1.5723(-4) 4.00 9.8187(-6) 4.00 6.1364(-7) t = 1 e 1.2074(-4) 4.00 7.5434(-6) 4.00 4.7145(-7) er 4.8103(-4) 3.0066(-5) 1.8791(-6) ex 1.2255(-4) 4.00 7.6526(-6) 4.00 4.7826(-7)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Second Test Problem -second order time stepping Similar results are shown for the non-homogeneous problem for 0 ≤ x, y ≤ 1 with ν = 1, with exact solution ψ = e−2x−ye−t.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

mesh 17 × 17 Rate 33 × 33 Rate 65 × 65 t = 0.25 e 5.5306(-7) 3.97 3.5412(-8) 3.98 2.2491(-10) er 2.4301(-7) 1.4728(-8) 9.1050(-10) ex 6.2534(-5) 3.90 4.1890(-8) 3.93 2.7576(-9) t = 0.5 e 4.2671(-8) 3.96 2.7421(-9) 3.98 1.7429(-10) er 2.4126(-7) 1.4644(-8) 9.0600(-10) ex 4.8421(-7) 3.90 3.2522(-8) 3.93 2.1389(-8) t = 0.75 e 3.3112(-8) 3.96 2.1259(-9) 3.97 1.3521(-10) er 2.3988(-7) 1.4577(-8) 9.0244(-10) ex 3.7539(-7) 3.87 2.5266(-8) 3.95 1.6605(-9) t = 1 e 2.5671(-8) 3.96 1.6494(-9) 3.97 1.0497(-10) er 2.3879(-7) 1.4525(-8) 8.9965(-10) ex 2.9132(-7) 3.89 1.9639(-8) 3.93 1.2900(-9)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The Driven Cavity The domain is Ω = [0, 1] × [0, 1] and the fluid is driven in the x−direction on the top section of the boundary (y = 1). Thus, u = 1, v = 0 for y = 1, and u = v = 0 for x = 0, x = 1 and y = 0.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The Driven Cavity - Re=7500, 10000, 4th order scheme

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 Velocity Components, Re=7500, 4th order scheme, T=560, mesh 65x65

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 Velocity Components, Re=10000, 4th order scheme, T=500, mesh 65x65

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Maximal streamfunction values for Re=10000, 4th order scheme

400 410 420 430 440 450 460 470 480 490 500 0.119 0.1195 0.12 0.1205 0.121 0.1215 0.122 0.1225 0.123 0.1235

(a) 400 < T < 500

2712 2713 2714 2715 2716 2717 2718 2719 2720 2721 2722 0.1225 0.1225 0.1225 0.1225 0.1225 0.1225 0.1226 0.1226 0.1226 0.1226 0.1226

(b) 2712 < T < 2722

Figure: Driven Cavity for Re = 10000 : Max streamfunction. Computations are done with N = 65, with ∆t = 1/90.

Pan and Glowinski, 2000.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Fourth-order time-stepping scheme Let U n , U n+1 be approximations to ∆ψ at times tn = n∆t and tn+1 = (n + 1)∆t. Given U n, we construct U n+1 via the following multi-step IMEX scheme of the form U i = U n − ∆t s

j=1 aC i,jC(U j) + ∆t s j=1 aD i,jD(U j), i = 0, ..., s

U n+1 = U n − ∆t s

i=1 bC i C(U i) + ∆t s i=1 bD i D(U i).

(35) The coefficients aC

i,j, bC i , and aD i,j ,bD i are given in Kennedy and

Carpenter, 2003. For this scheme s = 5, thus there are 5 intermediate terms and a final one. Kennedy and Carpenter, Appl. Numer. Math., 2003 Joint work with S. Gottlieb.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Fourth-order time-stepping scheme

Table: Explicit RK coefficients tableau for the convective term

ci aC i,1 aC i,2 aC i,3 aC i,4 aC i,5 aC i,6 1 2 1 2 83 250 13861 62500 6889 62500 31 50 − 116923316275 2393684061468 − 2731218467317 15368042101831 9408046702089 11113171139209 17 20 − 451086348788 2902428689909 − 2682348792572 7519795681897 12662868775082 11960479115383 3355817975965 11060851509271 1 647845179188 3216320057751 73281519250 8382639484533 552539513391 3454668386233 3354512671639 8306763924573 4040 17871 bC i 82889 524892 15625 83664 69875 102672 − 2260 8211 1 4 Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Fourth-order time-stepping scheme

Table: Implicit RK coefficients tableau for the viscous term

ci aD i,1 aD i,2 aD i,3 aD i,4 aD i,5 aD i,6 1 2 1 4 1 4 83 250 8611 62500 − 1743 31250 1 4 31 50 5012029 34652500 − 654441 2922500 174375 388108 1 4 17 20 15267082809 155376265600 − 71443401 120774400 730878875 902184768 2285395 8070912 1 4 1 82889 524892 15625 83664 69875 102672 − 2260 8211 1 4 bD i 82889 524892 15625 83664 69875 102672 − 2260 8211 1 4 Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical Results - high-order time-stepping Test problem ∂(∆ψ) ∂t + (∇⊥ψ) · ∇(∆ψ) = ν∆2ψ + f(x, y, t), for 0 ≤ x, y ≤ π with ν = 1, with exact solution ψ = −0.5e−2tsin(x)sin(y) The following table summarizes the error, e, the relative error, er, and the error in ψx in the l2 norm, where el2 = ψcomp − ψexactl2, er = el2/ψexactl2 (ex)l∞ = (ψx)comp − (ψx)exact∞.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

mesh 17 × 17 Rate 33 × 33 Rate 65 × 65 t = 0.3927 e 2.9389E-05 3.60 2.4222E-06 3.48 2.1655E-07 er 8.2063E-04 6.7647E-06 6.0474E-07 ex 2.3556E-05 3.41 2.2145E-06 3.22 2.3684E-07 t = 0.7854 e 1.3489E-05 3.78 9.8393E-07 3.67 7.7175E-08 er 8.2611E-05 6.0264E-06 4.7269E-07 ex 1.1070E-05 3.70 9.8627E-07 3.25 1.0363E-07 t = 1.1781 e 6.1469E-06 3.83 4.3337E-07 3.74 3.2571E-08 er 8.2569E-05 5.8267E-06 4.3754E-07 ex 5.0524E-06 3.50 4.4666E-07 3.26 4.6709E-08 t = 1.5708 e 2.8002E-06 3.73 1.9591E-07 3.75 1.4527E-08 er 8.2549E-05 5.7723E-06 4.2803E-07 ex 2.3034E-06 3.50 2.0327E-07 3.26 2.1229E-08

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

A High Order scheme for Irregular domains using 2D polynomials

M0(ih, jh)

Figure: Embedding of an elliptical domain in a Cartesian grid. Calculated nodes :

black circles. Exterior points : black squares. Edge Points: white circles.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains M1(−h1, h1) M2(0, h2) M3(h3, h3) M4(−h4, 0) M5(h5, 0) M6(−h6, −h6) M7(0, −h7) M8(h8, −h8) M0(0, 0)

[8] M. Ben-Artzi, I. Chorev, J.-P . Croisille and D. Fishelov, SINUM 2009. [9] M. Ben-Artzi, J.-P . Croisille and D. Fishelov, BGSiam, 2017.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

A Hermite-Lagrange interpolation problem in two dimensions The sixth-order polynomial PM0(x, y) is of the form P(x, y) =

19

• i=1

aili(x, y), (36)                              l1(x, y) = 1, l2(x, y) = x, l3(x, y) = x2, l4(x, y) = x3, l5(x, y) = x4, l6(x, y) = x5, l7(x, y) = y, l8(x, y) = y2, l9(x, y) = y3, l10(x, y) = y4, l11(x, y) = y5, l12(x, y) = xy, l13(x, y) = xy(x + y), l14(x, y) = xy(x − y), l15(x, y) = xy(x + y)2, l16(x, y) = xy(x − y)2, l17(x, y) = xy(x + y)3, l18(x, y) = xy(x − y)3, l19(x, y) = x2y2(x2 + y2). (37)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The rectangular 9 point stencil

Figure: In the case of a calculated interior point , the picture shows the 19-data.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Test cases in non convex domains-a flower domain We consider first the biharmonic problem in a non-convex domain. The boundary of the domain is given in polar coordinates by x(θ) = R(θ) cos(θ), y(θ) = R(θ) sin(θ), 0 ≤ θ < 2π (38) with R(θ) = 0.6 + 0.25 sin(7θ). The exact solution is ψ(x, y) = x2 + y2 + ex cos(y). The numerical results show outstanding accuracy even for very coarse grids.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Star shaped domains

Figure: Seven branches star shaped domain embedded in a 33 × 33 grid. Left: domain and grid. Right: approximate solution corresponding to ψ(x, y) = x2 + y2 + ex cos(y). • blue circles: boundary points, • green circles: edge points.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

7 and 9 branches star shaped domain: convergence rate

Figure: Linear regression of the convergence rate for ˜ ψ − ψ∞ and ˜ ψx − ψx∞ with ψ(x, y) = (x, y) = x2 + y2 + ex cos(y). The 6 points correspond to the 6 grids 10 × 10, 20 × 20, 30 × 30, 40 × 40, 50 × 50 and 60 × 60.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Biharmonic problem (19 branches) in star-shaped domain

Figure: 19 branches with a 100 × 100 grid. • Left: domain and grid. • Right: approximate solution Error values: ˜ ψ − ψ = 1.26 10−5, ˜ ψx − ψx = 3.39 10−4

.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

A High Order scheme for Irregular domains using 2D polynomials

Figure: Grid point in the ellipse: Left: 40 × 40 points. Right: 60 × 60 points.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

NS equations in an ellipse : computed solution and the error

Figure: Convergence rates for the ellipse with exact solution ψ(x, y, t) =

• (x2 + 4y2) − 1/4
• cos(t). : Left: Computed solution with 60 × 60
• points. Right: The error.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

NS equations in an ellipse : convergence rates

Figure: Convergence rates for the ellipse: Left: Li-Wang scheme,2003. Right: Our scheme.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

A High Order scheme for Irregular domains using 1D polynomials

Figure: Grid: ’+’ computational point, ’o’ eight neighbors of a computational point, ’x’ point too close to the boundary.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

A High Order scheme for Irregular domains using 1D polynomials Define a new coordinate system η = (x + y)/ √ 2, ξ = (y − x)/ √ 2. Thus, y = (η + ξ)/ √ 2, x = (η − ξ)/ √

• 2. By the chain rule,

ψηηηη = 1

4(ψxxxx + 4ψxxxy + 6ψxxyy + 4ψxyyy + ψyyyy)

ψξξξξ = 1

4(ψxxxx − 4ψxxxy + 6ψxxyy − 4ψxyyy + ψyyyy).

(39) Therefore, 2(ψηηηη + ψξξξξ) = ψxxxx + 6ψxxyy + ψyyyy. Thus, ∆2ψ = ψxxxx + 2ψxxyy + ψyyyy = 2

3(ψηηηη + ψξξξξ + ψxxxx + ψyyyy).

(40)

• A. Ditkowski, private communications

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The discrete convective for an irregular element The convective term is C(ψ) = ∇⊥ψ · ∇∆ψ = −(∂yψ) · ∂x(∆ψ) + (∂xψ) · ∂y(∆ψ). It may be written as C(ψ) = −(∂yψ) · (∂xxxψ + ∂xyyψ) + (∂xψ) · (∂xxyψ + ∂yyyψ). [10] D. Fishelov, Computers and Mathematics with Applications, 2017.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The discrete convective for an irregular element For the mixed third-order derivatives we have. ψηηη = 1 2 √ 2(ψxxx + 3ψxxy + 3ψxyy + ψyyy), ψξξξ = 1 2 √ 2(−ψxxx + 3ψxxy − 3ψxyy + ψyyy). Therefore, ψxxy = √ 2 3 (ψηηη + ψξξξ) − 1 3ψyyy, ψxyy = √ 2 3 (ψηηη − ψξξξ) − 1 3ψxxx. The convective term may be written using only pure derivatives by C(ψ) = −ψy· 2 3ψxxx+ √ 2 3 (ψηηη−ψξξξ)

• +ψx·

2 3ψyyy+ √ 2 3 (ψηηη+ψξξξ)

• .

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

The truncation error for ∂x, ∂4

x for an irregular element

Let Q(x) be the following polynomial with interpolating data Q(x) = a0 + a1(x − xi) + a2(x − xi)2 + a3(x − xi)3 + a4(x − xi)4, ψ(xwest, yj), ψ(xi, yj), ψ(xeast, yj), ψx(xwest, yj), ψx(xeast, yj). Then, the approximation ψx,i,j to ∂xψi,j has the form ψx,i,j + cx,p · ψx(xeast, yj) + cx,m · ψx(xwest, yj) = cp · ψ(xeast, yj) − cm · ψ(xwest, yj) − c · ψi,j. (41) The truncation errors for ψx and ¯ δ4

x for an irregular element satisfy

|(ψx)i,j − ∂xψ| ≤ Ch4ψ(5)L∞, |¯ δ4

xψi,j − ∂4 xψ| ≤ Chψ(5)L∞.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Numerical Results- Irregular Domains-Full Navier-Stokes Intersection of two non-concentric circles Ω = {(x, y)|(x − 0.4)2 + y2 < 0.5} ∩ {(x, y)|(x + 0.4)2 + y2 < 0.5} (42) ψ(x, y, t) =

1 64(0.81 − (x2 + y2)2)e−t in Ω

We resolve numerically                      ∂t∆ψ + ∇⊥ψ · ∇∆ψ − ∆2ψ = f(x, y, t), (x, y) ∈ Ω ψ(x, y, 0) =

1 64(0.81 − (x2 + y2)2),

(x, y) ∈ Ω ψ(x, y, t) =

1 64(0.81 − (x2 + y2)2)e−t,

(x, y) ∈ ∂Ω

∂ψ(x,y,t) ∂n

=

1 64 ∂((0.81−(x2+y2)2)e−t ∂n

, (x, y) ∈ ∂Ω. (43)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

mesh 11 × 11 Rate 21 × 21 Rate 41 × 41 e2 5.8018E-09 3.87 3.9712E-10 3.86 2.7436E-11 e∞ 1.1809E-08 4.20 7.25789E-10 3.98 4.6122E-11 (ex)2 2.1158E-08 4.30 1.0708E-09 3.86 7.3503E-11 (ex)∞ 3.7714E-08 4.15 2.1361E-09 3.94 1.3377E-10

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 0.005 0.01 0.015

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

• 2

2 4 6 x 10

• 11

Figure: Left: Approximation for ψ(x, y, t) =

1 64(0.81 − (x2 + y2)2)e−t. Right:

The error

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

ψ(x, y, t) = (1/64)e−t((x2 + y2)2 + cos(x) · cos(y)) on Ω We resolve numerically                      ∂t∆ψ + ∇⊥ψ · ∇∆ψ − ∆2ψ = f(x, y, t), (x, y) ∈ Ω ψ(x, y, 0) = (1/64)((x2 + y2)2 + cos(x) · cos(y)), (x, y) ∈ Ω ψ(x, y, t) = (1/64)e−t((x2 + y2)2 + cos(x) · cos(y)), (x, y) ∈ ∂Ω

∂ψ(x,y,t) ∂n

= ∂(1/64)e−t((x2+y2)2+cos(x)·cos(y))

∂n

, (x, y) ∈ ∂Ω. (44)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

mesh 11 × 11 Rate 21 × 21 Rate 41 × 41 e2 5.1468E-08 4.02 3.1644E-09 4.68 1.2311E-10 e∞ 1.0385E-07 3.94 6.7449E-09 4.80 2.4166E-10 (ex)2 3.1049E-07 4.28 1.5965E-08 4.61 6.5458E-10 (ex)∞ 7.1207E-07 4.25 3.7389E-08 3.85 2.5906E-09

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 0.01 0.02

• 1
• 0.5

0.5 1

• 1

1

• 1.5
• 1
• 0.5

0.5 1 1.5 2 2.5 x 10

• 10

Figure: Left: Approximation for ψ(x, y, t) = (1/64)e−t((x2 + y2)2 + cos(x) · cos(y)). Right: The error

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

ψ(x, y, t) = (1/64)e−t((x2 + y2)2 + ex cos(y)) Our aim is to recover ψ(x, y, t) from f(x, y, t). Thus, we resolve numerically                      ∂t∆ψ + ∇⊥ψ · ∇∆ψ − ∆2ψ = f(x, y, t), (x, y) ∈ Ω ψ(x, y, 0) = (1/64)((x2 + y2)2 + ex cos(y)), (x, y) ∈ Ω ψ(x, y, t) = (1/64)e−t((x2 + y2)2 + ex cos(y)), (x, y) ∈ ∂Ω

∂ψ(x,y,t) ∂n

= ∂(1/64)e−t((x2+y2)2+ex cos(y))

∂n

, (x, y) ∈ ∂Ω. (45)

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

mesh 11 × 11 Rate 21 × 21 Rate 41 × 41 e2 3.0809E-08 4.02 1.8993E-09 4.33 9.4105E-11 e∞ 9.6878E-08 4.21 5.2525E-09 4.25 2.7563E-10 (ex)2 2.8732E-07 4.17 1.5968E-08 4.16 8.9395E-10 (ex)∞ 5.6380E-07 4.28 2.8971E-08 3.63 2.3323E-09 Table 10: Compact scheme for Navier-Stokes equation with exact solution: ψ = (1/64)e−t((x2 + y2)2 + ex cos(y)) on Ω. We present e and ex, the l2 errors for the streamfunction and for ∂xψ. Here ∆t = 0.25h2 and t = 0.16.

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

• 0.01

0.01 0.02 0.03 0.04

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

• 3
• 2
• 1

1 x 10

• 10

Figure: Left: Approximation for ψ(x, y, t) = (1/64)e−t((x2 + y2)2 + ex cos(y)). Right: The error

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations

Navier-Stokes equations in streamfunction formulation Optimal convergence in 1D The 2D Navier-Stokes system Numerical Results A high order scheme for irregular domains

Summary

• 1. Navier-Stokes equations in streamfunction formulation
• 2. Optimal convergence of a compact fourth-order scheme in 1D
• 3. The time-dependent problem in 1D
• 4. Fourth order schemes for N-S equations in 2D regular and

irregular domains

Dalia Fishelov Afeka-Tel-Aviv Academic College of Engineering High-order compact schemes for the Navier-Stokes Equations