introduction to numerical schemes - technion dascal cs 236861 - tutorial 2 - introduction to...

236861 Numerical Geometry of Images

Tutorial 2

Introduction to numerical schemes

Lorina Dascal

Lorina Dascal CS 236861 - Tutorial 2 - Introduction to numerical schemes

Heat equation

The simple parabolic PDE

∂u

∂t= K

∂2u

∂2x

with the initial values

u(0, x) = u0(x)

and some boundary conditions is called the (one-dimensional) heatequation.


I This equation describes the thermal energy transport in a 1Drod, where u(t, x) describes the temperature at a point x attime t and K denotes the thermal conductivity constant.

I In 2D it describes the effect of defocusing on an image inoptics

I The solution is given by convolution of the initial data with aGaussian kernel:

u(x , t) = G (x , t) ∗ u0(x) =

∫

RG (x̄ , t)u0(x − x̄)dx̄


Boundary conditions

Dirichlet:

u(t, 0) = a u(t, 1) = b

Neumann:

ux(t, 0) = a ux(t, 1) = b

Mixed boundary conditions:

u(t, 0) = a, ux(t, 1) = b.

Periodic:

u(t, 0−) = u(t, 1+)


Discretization of the heat equation

Replace the continuous system of coordinates (t, x) by a discretegrid (n, m) = (n∆t,m∆x), and the continuous function u(t, x) bya discrete version un

m = u(n∆t, m∆x).

Figure: The grid


Finite differences

1. Replace the first-order time derivative ut(t, x) by a forwardfinite difference in time

D+t un

m =un+1m − un

m

∆t≈ ut(t).

2. Replace the second-order space derivative uxx(t, x) by acentral difference in space

D0xxu

nm =

unm+1 − 2un

m + unm−1

(∆x)2≈ uxx(x).


Finite differences’ motivation

From the Taylor expansion,

u(t + ∆t) = u(t) + ut(t)∆t +1

2utt(t)(∆t)2 + ...

we obtain

ut(t) =u(t + ∆t)− u(t)

∆t− 1

2utt(t)∆t + ...

=u(t + ∆t)− u(t)

∆t+ O(∆t).

This first-order approximation of the first-order derivative is calledthe forward finite-difference approximation and is denoted byD+

t unm.


Finite differences

Forward difference

D+t un

m =un+1m − un

m

∆t≈ ut(t) + O(∆t).

In the same manner, backward difference can be defined

D−t un

m =unm − un−1

m

∆t≈ ut(t) + O(∆t).

To approximate a second-order derivative, use the centraldifference approximation

D0xxu

nm =

unm+1 − 2un

m + unm−1

(∆x)2≈ uxx(x) + O((∆x)2).

D0xxu

nm = D−

x D+x un

m.


The discrete heat equation

The continuous equation ut(t, x) = Kuxx(t, x) is replaced by

un+1m − un

m

∆t=

unm+1 − 2un

m + unm−1

(∆x)2

or

un+1m = un

m +∆t

(∆x)2K

(unm+1 − 2un

m + unm−1

),

This scheme is explicit (called also Euler scheme) and very simpleto implement.


I We denote the continuous solution by U = u(t, x) and thediscrete solution by u = un

m.

I We denote the continuous PDE operator L = ∂t − K∂xx .

I We denote the discrete operator L∆t,∆x = D+t − KD0

xx .


Stability and Convergence

1. Convergence: the discrete solution converges to thecontinuous solution

lim∆t,∆x→0

‖u − U‖ .

in certain norm.

2. Consistency: the discrete difference operator solutionconverges to the continuous solution

lim∆t,∆x→0

‖L(u)− L∆t,∆x(u) ‖ = 0

for every bounded u.


Stability

3. Numerical stability: noise from initial conditions, numericalerrors, etc. is not amplified.The numerical scheme is un+1 = Nun

Formally, stability means

∀N > 0 ∃C (N) > 0 s.t. ∀n ≤ N ‖N n‖ ≤ C (N).

Lax’s equivalence theorem: Given a properly posed initial valueproblem and a finite-difference approximation to it that satisfiesthe consistency condition, stability is the necessary and sufficientcondition for convergence.

consistency + stability ⇔ convergence


0 10 20 30 40 50 60 70−8

−6

−4

−2

0

2

4

6

8Initial data

0 10 20 30 40 50 60 70−5

−4

−3

−2

−1

0

1

2

3

4

5 Solution to 1−D heat equation

dt=0.0008

0 10 20 30 40 50 60 70−5

−4

−3

−2

−1

0

1

2

3

4

5Solurion to 1−D heat eq

dt=0.004

0 10 20 30 40 50 60 70−4

−3

−2

−1

0

1

2

3

4x 10

70

dt=0.006

Figure: Computation of the discrete heat equation with Neumannboundary conditions. (h = 0.1) Top left: Initial data. Top right: stablesolution of the heat equation conditions ∆t = 0.0008. Bottom left:stable solution of the heat equation for ∆t = 0.004. Bottom right:unstable solution (∆t = 0.006).


CFL condition for stability of the discrete heat equation

Denote r = K ∆t(∆x)2

. The finite difference scheme is

un+1m = un

m + r(unm+1 − 2un

m + unm−1

)

= (1− 2r)unm + run

m+1 + runm−1

If 1− 2r > 0, we can write:

un+1m ≤ (1− 2r)max

munm + r max

munm + r max

munm

Then un+1m ≤ max

munm(1− 2r + 2r) = max

munm.

(maximum principle property)


Observation: Maximum principle implies stability of the scheme.

Lemma(Maximum principle)If K ∆t

(∆x)2≤ 1/2, then

minm

u0m ≤ un

m ≤ maxm

u0m.


Implicit finite difference schemes for the heat equation

un+1m − un

m

∆t= K

un+1m+1 − 2un+1

m + un+1m−1

(∆x)2

I Main advantage of implicit schemes: are unconditionallystable (i.e. stable for all time-steps and thus we can take largestep times).(I −∆tA)Un+1 = Un.

I Requires solving a linear system of equations. The matrix(I −∆tA) is tridiagonal and can to be inverted using theThomas algorithm (we will describe it in detail in a futuretutorial).


The 2D heat equation. Numerical scheme.

ut = K (uxx + uyy )

un+1m,p = un

m,p + K( ∆t

(∆x)2(un

m+1,p − 2unm,p + un

m−1,p)+

+∆t

(∆y)2(un

m,p+1 − 2unm,p + un

m,p−1)).

The CFL condition:

K( ∆t

(∆x)2+

∆t

(∆y)2

)≤ 1

2.

If ∆x = ∆y , then the CFL condition is:

K∆t

(∆x)2≤ 1

4.


The 2D heat equation. Numerical example

Figure: Application of the heat eq. on a gray-scale image.


introduction to numerical schemes - technion dascal cs 236861 - tutorial 2 - introduction to...

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