proposal presentation

Introduction Elliptic problem Parabolic problem Proposed Work Conclusion

Optimal Finite Difference Grids for Elliptic andParabolic PDEs with Applications

Oleksiy VarfolomiyevAdvisor Prof. Michael Siegel

Co-Advisor Prof. Michael Booty

NJIT, May 15, 2012


Outline

1 IntroductionMotivation

2 Elliptic problemProblem FormulationDiscretization and NtD mapApproximation ErrorGrids and Numerical ResultsNtD map for nonuniformly spaced boundary data

3 Parabolic problemProblem FormulationDiscretizationBenchmarks

4 Proposed Work

5 Conclusion


Motivation

Motivation

Problem

Accurate and efficient computation of the DtN (NtD) maps

Applications of interest

1 Water waves: DtN map is used to compute the normalinterface speed

2 Crystal growth: DtN map is used to track the crystal-meltinterface

3 Surface with soluble surfactant: DtN map is used to resolvesurfactant concentration gradient


Elliptic Problem


Problem Formulation

Model Elliptic Problem Formulation 1

Laplace equation on a semi-infinite strip

−∂2w(x , y)

∂y2− ∂2w(x , y)

∂x2= 0, (x , y) ∈ [0,∞)× [0, 1], (1)

∂w

∂x(0, y) = −ϕ(y), y ∈ [0, 1], (2)

w |x=∞ = 0, (3)

w(x , 0) = 0, w(x , 1) = 0, x ∈ [0,∞). (4)

Our goal is to accurately resolve the Dirichlet data w(0, y)

1V. Druskin


Problem Formulation

BC in the Fourier space

φ(m)(y) =m∑i=1

ai sin(iπy), (5)

Fourier representation of the solution

w (m)(x , y) =m∑i=1

wi (x) sin(√λiy), (6)

w (m)(0, y) =m∑i=1

wi (0) sin(√λiy) =

m∑i=1

ai f (λi ) sin(√λiy), (7)

λi = (iπ)2, f (λ) = λ−12 is the impedance function


Discretization and NtD map

Discretization

Fourier transform (1) in y

λk wk(x)− ∂2wk(x)

∂x2= 0, k = 1, . . . ,m (8)

∂wk(0)

∂x= −ϕk , k = 1, . . . ,m, (9)

where λk = (kπ)2.

Discretize in x

λwi −1

hi

(wi+1 − wi

hi− wi − wi−1

hi−1

)= 0, i = 2, . . . , n, (10)

λw1 −1

h1

(w2 − w1

h1

)= − 1

h1

(w1 − w0

h0

)= Φ (11)

where wi = wk(xi ), Φ = ϕk , (w1 − w0)/h0 is set to Φ and λ = λk .



Discretization

Fourier transform (1) in y

λk wk(x)− ∂2wk(x)

∂x2= 0, k = 1, . . . ,m (8)

∂wk(0)

∂x= −ϕk , k = 1, . . . ,m, (9)

where λk = (kπ)2.Discretize in x

λwi −1

hi

(wi+1 − wi

hi− wi − wi−1

hi−1

)= 0, i = 2, . . . , n, (10)

λw1 −1

h1

(w2 − w1

h1

)= − 1

h1

(w1 − w0

h0

)= Φ (11)

where wi = wk(xi ), Φ = ϕk , (w1 − w0)/h0 is set to Φ and λ = λk .



Discrete NtD map

Discretization gives the NtD map in the form of continued fraction

w1 =1

h1λ+ 1h1+

1

h2λ+···+1

hn−1+1

hnλ+1hn

Φ ≡ Rn(λ)Φ (12)

Thus the problem of the grid optimization with respect to theNeumann-to-Dirichlet map error can be reduced to the problem ofthe uniform rational approximation of the inverse square root.



F - Fourier transform in y , F−1 - inverse Fourier transform in y

NtD Map

Given Neumann data ∂w∂x (0, y) = −ϕ(y), y ∈ [0, 1]

w1 = F−1RnFϕ, (13)

DtN Map

Given Dirichlet data w(0, y) = ψ(y), y ∈ [0, 1]

φ = −F−1R−1n Fψ (14)


Approximation Error

Approximation Error

L2 error of the semidiscrete solution at x = 0

en = ||w (m)(0, y)− w(n)1 (y)||L2[0,1] ≤ ||ϕ||L2[0,1] max

λ∈[π2,(mπ)2]|fn(λ)− λ−

12 |

En(λ) = maxλ∈[λmin,λmax ]

∣∣∣fn(λ)− λ−12

∣∣∣ = O

[exp

(π2n

logλminλmax

)]

The described special choice of the discretization grid stepsprovides a spectral convergence order of the solution at theboundary.


Approximation Error

Relative Error Plot

Relative error E (λ) =∣∣∣Pk (λ)−λ−1/2

λ−1/2

∣∣∣

0.1 1 10 100 1000 104

105

10-14

10-12

10-10

10-8

10-6

10-4

0.01

Λ

rela

tiv

eerr

or

1 10 100 1000 104

10-13

10-10

10-7

10-4

Figure: Relative Error in the approximation of the inverse square root,k = 16, λ ∈ [1, 10000] (left), λ ∈ [1, 1000] (right)


Grids and Numerical Results

Optimal geometric grids

α = eπ/√k h1 = e−

√kπ

hi = h1αi−1 = eπ(i−k−1)/

√k

h1 = h1/(1 +√α)

hi =√

hihi−1

Nonoptimal geometric grids

h1 = O(hy ) h1 =hy

1 +√α

100

101

102

103

104

105

106

107

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

fk(λ) − λ

−1/2

err

or

frequency

100

101

102

103

104

105

106

107

10−8

10−7

10−6

10−5

10−4

10−3

fk(λ) − λ

−1/2

err

or

frequency


Grids and Numerical Results

Numerical Results

We discretize second derivatives in y using a second order,centered finite difference scheme. The resulting system matrix issparse-banded.

Example with Neumann data ϕ(y) = sin(πy)

(a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−11 Abs. error in approximation of Dir BC

(b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−4 Abs. error in approximation of Dir BC

(c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Abs. error for node x2

Figure: (a) Error E (y) = |w(0, y)− F−1RnFϕ(y)| ; (b) errorE (y) = |w(0, y)− w1(y)|; (c) error obtained by the standard five-pointfinite difference scheme. (n = 16)


NtD map for nonuniformly spaced boundary data

Proposed work: NtD map for nonuniformly spacedboundary data

Semi-discrete system

Aw = (ϕ, 0)T , w = (w1, ...,wn)T , (15)

A =

∂2

∂y2 + 1h1h1

− 1h1h1

. . . . . .

− 1hihi−1

∂2

∂y2 + 1hihi

+ 1hihi−1

− 1hihi−1

. . .

......

.... . .

This system can be solved using GMRES


NtD map for nonuniformly spaced boundary data

Computation of the derivatives 2

Assume y = y(α) and yj = y(αj), where αj are uniformly spacedThe derivative is computed as

dw

dy(yj) =

dw

dα(αj)

1dydα(αj)

which can be computed for α = αj by the FFT

This method will be applied to compute Hilbert and Riesztransforms for nonuniformly spaced points in O(N logN)

operations, with spectral accuracy

2C. Muratov


Parabolic Problem


Problem Formulation

Model Parabolic Problem Formulation 3

Heat equation on a semi-infinite strip

∂u(x , t)

∂t=∂2u(x , t)

∂x2, (x , t) ∈ [0,∞)× [0,T ], (16)

u(x , 0) = 0, x ∈ [0,∞) (17)

u(x , t)|x=∞ = 0, u(0, t) = g(t), t ∈ [0,T ] (18)

3M. Booty


Problem Formulation

Laplace transform the equation (16) in time, we get

λu(x , t) =∂2u(x , t)

∂x2, (19)

Therefore

u(x , t) = u(0, t)e−λ1/2x = g(x)e−λ

1/2x (20)

and∂u

∂x(x = 0) = −

√λg(x = 0), (21)


Discretization

Discretizing in real space in x

u1 − u0h1/2

− u−1/2 − h0∂u0∂t

= 0

1

hi

(ui+1 − uihi+1/2

− ui − ui−1hi−1/2

)− ∂ui∂t

= 0, i = 1, . . . , n − 1 (22)

un = 0

u0(t), ∂u0(t)∂t are known from the Dirichlet BC and the initial

data ui (t = 0) are known

solve (22) to update ui , i = 1, . . . , n − 1 to the time t = 4t

the process is repeated to obtain the Neumann data un−1/2 atdiscrete time tn = n4t


Benchmarks

Benchmark 1: const BC

∂u(x , t)

∂t=∂2u(x , t)

∂x2, (x , t) ∈ [0,∞)× [0,T ], (23)

u(x , 0) = 0, x ∈ [0,∞) (24)

u(x , t)|x=∞ = 0, u(0, t) = 1, t ∈ [0,T ] (25)

Using the Laplace transform we get

u(x , t) = erfc

(x

2√t

)=

2√π

∫ ∞x

2√t

e−u2du (26)

ux(0, t) = − 1√πt

(27)


Benchmarks

DtN Error

a)0 2 4 6 8 10 12 14 16 18 20

−8

−6

−4

−2

0

2

4

6

8

10

12x 10

−3

Numerical Error of ux(0,t)

Mt = 1541877, Nx = 16, dt = 1.2971e−05

0.0042416 < t < 20 b)0 2 4 6 8 10 12 14 16 18 20

−12

−10

−8

−6

−4

−2

0

2x 10

−3


Mt = 1541877, Nx = 16, dt = 1.2971e−05

0.0042027 < t < 20

c)0 2 4 6 8 10 12 14 16 18 20

−12

−10

−8

−6

−4

−2

0

2x 10

−3


Mt = 5553, Nx = 16, dt = 0.0036016

0.18008 < t < 19.9994

Figure: Error of ux(0, t) for the BC u(0, t) = 1 with k = 16. a) FE intime with 4t = 4x2, b) BE in time 4t = 4x2, c) BE in time 4t = 4x


Benchmarks

Benchmark 2: harmonic BC

∂u(x , t)

∂t=∂2u(x , t)

∂x2, (x , t) ∈ [0,∞)× [0,T ], (28)

u(x , 0) = 0, x ∈ [0,∞) (29)

u(x , t)|x=∞ = 0, u(0, t) = b sin (ωt), t ∈ [0,T ] (30)

u(x , t) = b e−√

ω2xsin

(ωt −

√ω

2x

)+

bω

π

∫ ∞0

e−ut sin(x√u)

u2 + ω2du

ux(0, t) = −b√ω

2(sin (ωt) + cos (ωt))+

b

πωt32

+O

(1

t52

), as t →∞


Benchmarks

DtN Error

a) b)

c)

Figure: Error of ux(0, t) for the boundary condition u(0, t) = sin(t) a)with k = 4, b) with k = 8 and c) with k = 16


Benchmarks

Benchmark 3: Diurnal Earth Heating

∂u(x , t)

∂t=∂2u(x , t)

∂x2, (x , t) ∈ [0,∞)× [0,T ], (31)

u(x , 0) = 0, x ∈ [0,∞) (32)

u(0, t) = 1 + b sin (ωt), t ∈ [0,T ] (33)

ux(0, t) ≈ − 1√πt−b

√ω

2(sin (ωt) + cos (ωt))+

b

πωt32

, as t →∞


Benchmarks

DtN Error

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0


Mt = 1541877, Nx = 16, dt = 1.2971e−05

1.3636 < t < 20

Figure: Error of ux(0, t) for the boundary conditionu(0, t) = 1 + sin(t), k = 16


Proposed work: Crystal Growth Problem4

Infinite melt cooled below its freezing temperature. Two-phaseflow with the moving interface separating the crystal from themelt. We wish to model the evolution of the crystal-melt interface.

Mathematical model for the problem

The diffusion equation in the melt

∂T

∂t= D∇2T , (34)

where D is constant thermal diffusivity.The specified temperature in the melt far from the interface

T → T∞ as y →∞ (35)

4M. Kunka


The temperature at the interface with the parametrizationy = yi (x , t) is given by the Gibbs-Thomson relation

T = Tm(1− γκ) on y = yi (x , t), (36)

where κ - interface curvature and γ - capillary length thatcharacterizes the surface tension.And the kinematic condition relating the heat flux and the velocityof the moving interface

cD∂T

∂n= −Lvn on y = yi (x , t), (37)

vn - the normal velocity of the interface, L - the latent heat and c -specific heat.


We propose to investigate the crystal growth problem in theless well studied limit of large Peclet number. In this limit, weexpect a boundary layer adjacent to the moving interfacewhere the temperature gradients are large.

A singular perturbation analysis will be performed to derive aleading order equation governing the temperature in the layer.

We propose to efficiently solve this equation with highaccuracy using optimal grids.


Conclusion

We showed that the described special choice of thediscretization grid steps provides a spectral convergence orderof the solution at the boundary.

This method of computing the Riesz transform will be appliedto a new numerical method of Ambrose and Siegel forremoving the stiffness from boundary integral calculationswith surface tension.

We propose to efficiently solve the equation for the crystalgrowth problem in the limit of large Peclet number with highaccuracy using optimal grids.

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