Alternating Directions Implicit ADI Method

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Alternating Directions Implicit (ADI) Method

• We would like to have an unconditionally stable
  scheme
• that involves linear systems with a tridiagonal
  matrix!

Operators A may represent, for example, numerical
derivatives
Using the trapezoide rule we can write
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Alternating Directions Implicit (ADI) Method
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Alternating Directions Implicit (ADI) Method
Rearrange the two equations
These equations are equivalent to (if A1and A2
commute)
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Alternating Directions Implicit (ADI) Method
Back to our 2D heat equation
The operator A1 and A2 correspond to numerical
differences. Using central differences formulas
1st stage
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Alternating Directions Implicit (ADI) Method
Only the x-index varies in the left side ----gt
implicit in the x-direction only
Fixed y-index j Vector un1/2(, j) can be
found by solving the tridiagonal system
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Alternating Directions Implicit (ADI) Method
Where A is tridiagonal
and b is a Nx-dimensional vector with components
Boundary conditions have yet not been
incorporated!!
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Alternating Directions Implicit (ADI) Method
do j 1, Ny do i 1, Nx b(i) (1-2sy)
ui,j. end do solve Aunew(,j) b end do
Stage 1 of the ADI method is then implemented in
a loop over the y-direction
At this stage a tridiagonal system of dimension
Nx is solved Ny times
do i 1, Nx do j 1, Ny b(j) (1-2sx)
unew. end do solve Au(i,) b end do
Stage 2 of the ADI is implemented in a loop over
the x-direction
Stage 1 followed by Stage 2 advances the solution
with a ?t step in time, overwriting u. Unew is
only an intermediate stage.
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von Neumann Stability Analysis
The stability of a numerical method can be
studied (before even writing a program!) by the
von Neumann Method.
The Fourier transform of a function u(x) is
defined as
The inverse of this Fourier transform is a unique
representation of the function u as a
superposition of waves
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von Neumann Stability Analysis
The von Neumann stability analysis checks how the
Fourier modes evolve from one timestep to the
next.
Consider a Trial Solution (one Fourier mode out
of the full solution) at a given point x.
k is the wave number U(t) is the complex amplitude
In discretized form, we get
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von Neumann Stability Analysis
Advancing the solution by one time step
where g Un1/Un is called the amplification
factor
If g gt 1 then the solution grows in amplitude
and becomes unstable. If g lt 1 the solution
is dampened.
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von Neumann Stability Analysis
Strategy of von Neumann analysis is
I) Insert a trial solution into the numerical
scheme II) Solve for the amplification factor,
g, in terms of the grid spacing, h, and the
timestep, ?.
If g gt 1 then the solution grows in amplitude
and becomes unstable. If g lt 1 the solution
is dampened.
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von Neumann Stability Analysis
Stability of the 1-D explicit Diffusion Equation
We can carry out a von Neumann analysis of the
stability by inserting the trial solution
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von Neumann Stability Analysis
Stability of the 1-D explicit Diffusion Equation
For stability the amplification factor must
satisfy
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von Neumann Stability Analysis
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von Neumann Stability Analysis
Stability of the 1-D Implicit Diffusion Equation
We can carry out a von Neumann analysis of the
stability by inserting the trial solution
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von Neumann Stability Analysis
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Hyperbolic PDEs
One of the most important hyperbolic PDEs is the
wave equation
Where A(x, t) is the amplitude and c is the wave
speed.
The wave equation is however not the simplest
possible hyperbolic equation.
In order to understand the numerical methods
necessary for hyperbolic equations, it is best to
start with the easiest, nontrivial equation
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The Linear Advection Equation
One of the simplest hyperbolic PDEs is the linear
advection equation
Physically this equation describers the evolution
of a scalar field, a(x,t), carried along by a
flow of constant velocity, c.
An analytical solution of the advection equation
can be found by looking at the characteristics of
the equation.
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The Linear Advection Equation
The total time derivative of a(x,t) along any
curve x(t) is given by
This implies that along any curve which satisfies
The quantity a(x,t) is constant.
This curve is called the characteristic curve
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The Linear Advection Equation
So in our case of a constant velocity, c, we have
where x0 is a constant of integration (initial
condition)
Therefore our solution is
Any initial shape will move with velocity c along
x.