Lecture 6 Elliptic Eq Ns Linear System
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Transcript of Lecture 6 Elliptic Eq Ns Linear System
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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