Lecture 6 Elliptic Eq Ns Linear System

8/22/2019 Lecture 6 Elliptic Eq Ns Linear System

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AML 811

Lecture 6

Elliptic Equations

Solving a Linear System of Equations


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Recap of Lecture 5 : Stability of

discretizations If the PDE is stable, i.e. its solution remains bounded, then we

need that the finite differnce solution should also be stable

Some methods for analyzing stability1. Discrete perturbation method

2. Von-Neumann analysis

3. Matrix method

Stability analysis can normally only be done for linearequations. For non-linear equations, we linearize locally beforeanalysis and hence, only linear stability can be judged.

Experience indicates that linear stability normally means non-linear stability as well.


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Recap: Explicit FTCS for the diffusion

equation x = 0.2; t = 0.004; =1

2

2

x

u

t

u

=

FTCS

1=


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Explicit FTCS for the diffusion equation

Finer grid : x = 0.1; t = 0.004

FTCS2

2

x

u

t

u

=


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Explicit FTCS for the diffusion equation

Finer grid : x = 0.05; t = 0.004

FTCS2

2

x

u

t

u

=

Solution quickly

unstable. Why?


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Recap: Summary of Von-Neumann Analysis

Step 1 : Fourier Decomposition:

Assume that solution is composed of a sum of waves of

the form

Step 2: Obtain evolution equation for the amplitude

Substitute Fourier decomposition in the original Finite

Difference equation and write it in the formnn GUU =+1

Soln at grid

point i, at time

step n

Amplitude of

Fourier waveat n

Wave number

Gain oramplification

factor

Ref Computational Fluid Dynamics 1 : Hoffmann and Chiang: pgs 124-25


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Recap: Summary of Von-Neumann Analysis

Step 3 : Find region of stability

Find the conditions on x, t under which the amplitude of the wavewill be stable, i.e. not grow. This will happen if

Stability analysis shows that FTCS for the wave equationis stable only ift =0 i.e. FTCS is never stable for the

wave equation: Unconditionally unstable

FTCS for the diffusion equation

Stable only for2

sin41 2 dG =

2

12

=

x

td

Ref: Computational Fluid Dynamics -1 : Hoffmann and Chiang, pgs 124-25


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Recap: The Explicit Approach

Solution at each point in the next time step

computed from known values at previous

time step(s)Stencil

Example:


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Recap: The Implicit Approach

Solution at a certain grid point for a time step dependson values at other grid points in the same time step

A system of equations has to be solved to

simultaneously obtain the solution at all grid points This is computationally more expensive than the explicit

method as we have to solve a system of equations at

each time step. So, are there are any advantages?

Example:


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Recap: Implicit FTCS for the diffusion

equation Von-Neumann analysis shows

allfor1

2cos41

12

+

= Gd

G

Since there is no restriction on time step,unlike the explicit scheme, unconditional

stability allows us to take much larger timesteps


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Recap: FTCS for diffusion equation

These terms are computed at

time level n : known

Explicit FTCS

Implicit FTCS

These terms are computed at

time level n + 1 : unknown


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Recap: Other Schemes for the diffusion

equation

DuFort-Frankel

Explicit

Crank-Nicholson

Implicit


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Recap: Summary of Lecture 5

Even if a numerical scheme is consistent, it might not bestable for all choices of the grid spacing and time step.

Von Neumann stability analysis can be used to analyze

the linear stability of a numerical scheme Implicit schemes are more stable (many times

unconditionally stable) than explicit schemes and hence,

allow larger time steps. However, they involve greatercomputational expense per computational time step

Beta formulation for the 1D diffusion equation


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Elliptic Equations

Recall Have no real characteristics

Equilibrium problems. Oftenrepresent steady state of someparabolic problem. Example:Steady state heat-conduction

Boundary value problems(BVP) : Value of function onthe interior of the domain iscompletely determined by

values on the boundary. Alsoknown as jury problems forthe same reason.


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Examples of Elliptic Equations

Steady state heat conduction with isotropic material

02 = T

02 =

Streamfunction for 2D, incompressible, inviscid,

irrotational flow Laplaces Equation02 =

Creeping flow (very low speed flow) driven purely bygravity or other body forces

fur

r

=2

Poisson Equation

),,(2 zyxf=


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A simple elliptic problem 1D steady-

state heat conduction

02

2

=dx

TdBCs

LxTT

xTT

b

a

==

==

@

0@

Dirichlet

BCs

To numerically solve this, first discretize the 1D

domain into N points. Greater N => greateraccuracy

x = 0

i = 1x = L

i = N


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Dirichlet (value of function) and Neumann

(derivative of function) BCs

LxTT

xTT

b

a

==

==

@

0@

LxTT

xT

b

x

==

==

@

0@0

Neumann

BC

Dirichlet

BC

=

b

a

T

T

T

T

T

T

T

T

0

0

0

0

100000

121000

012100

001210

000121

000001

6

5

4

3

2

1

=

bTT

T

T

T

T

T

0

0

0

0

0

100000

121000

012100

001210

000121

000011

6

5

4

3

2

1

How can we solve such equations?


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Direct vs Iterative Methods Direct Methods

Solution found at one shot

Give exact solution (to precision of machine)

Example: Cramers rule, Gauss elimination

Extremely expensive O(N!) for Cramerrule

O(N3) for Gauss elimination

Iterative methods

Solution found by iteration. Gives approximate solution

Example: Jacobi iteration, Gauss-Seidel, SOR, Conjugate-Gradient

Work well forsparse linear systems: Systems which have lots ofzeros. These systems arise naturally for PDE approximations

Often, paradoxically, simpler to implement than direct methods


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A 2D Diffusion problem

BCs

No of unknowns = ?


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A 2D Diffusion problem

BCs

No of unknowns = 16


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A 2D Diffusion problem Five Point Stencil

Five Point stencil

Second order accurate


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A 2D Diffusion problem Nine Point Stencil

Nine Point stencilFourth order accurate


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System of equations for the second order

method

Note that the corner points never

enter into the equation


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System of equations for the second order

method

Sparse, band pentadiagonal system of equations.


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Iterative method for solving system of

equations: Jacobi method

Note that no matrix

entries are really

stored anywhere


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Iterative method for solving system of

equations: Jacobi method Start with initial assumed guess for solution

(except at the Dirichlet boundaries) u0

Update the values at all unknown pointsusing the equation for the diagonal term. For

Laplaces equation this comes to

Repeat iterations till successive iterations areclose enough


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Jacobi vs FTCS

FTCS

Jacobi

Jacobi solves the equilibrium problem as ifit is the steady-statesolution to the corresponding parabolic problem

This generally results in slower convergence of the iteration andsince we are not interested in the actual parabolic problem, wecan get faster solutions by modifying the Jacobi method


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Point Gauss-Seidel method

Start with initial assumed guess for solution

(except at the Dirichlet boundaries) u0

Update using the newest values available at

each grid point

G S id l i i


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Gauss-Seidel iterationUn-updated values

Updated values

S f L 6


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Summary of Lecture 6

Discretization of elliptic equations results in asystem ofsparse linear equations

Direct methods are often more expensivethan iterative methods for sparse linear

systems Jacobi for Laplace behaves similarly to the

FTCS method for the corresponding

parabolic problem Gauss-Seidel is a method that generally

converges faster than Jacobi

Lecture 6 Elliptic Eq Ns Linear System

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