Galerkin Method

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Galerkin Method
Main Idea Represent the solution in terms of
basis functions, coefficients
will be function of time PDE-gt ODE for
time-evolution coefficients (solvable e.g. with
Runge Kutta)
L is differential operator, u is dependent
variable and f is specified forcing function, BC
at xa and altxltb
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Galerkin Method
Error
Choose an(t) such that eN is orthogonal to each
basis function
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Galerkin Method
The above N algebraic equations can be used to
solve the N unknown coefficients a(t)
Spectral method
Finite element method
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Galerkin Method Example again diffusion equation
Time-discretization
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Galerkin Method Example again diffusion equation
Approximate
Example K1, n10, ?t0.1
Sign alternating solution Rubbish
Approximate
Exact
Numerical instability has to be avoided but
how????
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Von Neumann stability
A solution to a linear equation can be expressed
in terms of a Fourier series, where each harmonic
solution is also a solution We can test the
stability of a single harmonic solution
stability of all harmonics can then be a
necessary conditions for stability of the
scheme assume at some time (t0) that 1.) A
Fourier expansition of the initial field f(x) can
be made 2.) A separation of time and space
variables can be made, such that at time t a
simple term in the Fourier series is
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Von Neumann stability
Ensures numerical stability
4. Requiring
Is required physically
5. Oftentimes
Example 1-d diffusion equation, again and again
and again
Assume forward differencing in time and finite
differencing in space
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Von Neumann stability
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Solving partial differential equations
Hyperbolic wave equation Initial value (Cauchy)
problem
parabolic diffusion equation Initial value
problem
elliptic Poisson equation Boundary value problem
From a numerical point of this distinction is not
too important
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Solving PDEs
Initial value problem, defined by the anwers to
the following questions - What are the dependent
variables to be propagated in time? - What is the
time-evolution equation for each variable? - What
is the highest time derivative for each
variable? - What boundary conditions govern the
evolution in time? (e.g. Dirichlet Bcs
boundary values as function of time, Neumann
conditions specify normal gradient at the
boundary - Most crucial aspect is the stability
of the numerical scheme!!!
Boundary value problem, defined by the anwers to
the following questions - What are the
variables? - What equations are satisfied in the
interior of the domain? - What equations are
satisfied by points on the boundary region of
interest? - Most schemes are pretty stable - BC
must be satisifed simultanously -gt linear matrix
equations
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Example 1-d wave equation
Characteristics is location of constant phase in
x-t continuum
Dispersion-free movement of initial
wave-package Initial value problem, perturbation
moving to the left
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Example 1-d wave equation
Search for equation whose solution can propagate
both to the left and right
Classical wave equation
With
Solution of
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Example 1-d wave equation
Traveling wave moving to positive x (to the
right) Characteristics tx/c
Discretization xm?x, m0,1,2,.....
tn?t, m0,1,2,..... u(x,t)
u(m?x,n?t)umn Using forward differences in
time and centered in space (FTCS)
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Example 1-d wave equation Stability
P
Unconditionally unstable scheme
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Example 1-d wave equation Stability of FCTS
FTCS
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Example 1-d wave equation
Discretization xm?x, m0,1,2,.....
tn?t, m0,1,2,..... u(x,t)
u(m?x,n?t)Cmn Using centered differences
in time and space CTCS
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Example 1 d wave equation
Called leap-frog scheme, doesn't work for the
first two time steps use Euler forward (forward
in time, centered in space, FTCS) for first two
steps
T
N1
N
N-1
M
X
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Example 1 d wave equation
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Example 1 d wave equation
Stable scheme
If
If
Unstable scheme
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Example 1 d wave equation
For any k, stable scheme
Courant-Friedrichs-Lewy (CFL) stability criterion
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Example 1 d wave equation
CTCS
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Example 1 d wave equation
Mathematic problems with CTCS computational
dispersion and parasitic waves Physical problem
with CTCS negative values, bad for concentration
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CFL criterion
T
Means no amplitude loss But phase error
possible gt computational dispersion
Unstable
Stable
Stable
1/c
X
High velocities means small time-steps Characteri
stics of true equation have to lie within cone
of influence of CTCS scheme
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CFL criterion
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Example 1 d wave equation
Using uncentered differences in time and space
UCTCS
Unstable
Stable, but damping of amplitude
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Example 1 d wave equation
UCTCS
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Numerical diffusion in Euler backward scheme
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Numerical diffusion in Euler backward scheme
Numerical diffusion enters the advection equation
with equ. Diffusion coefficient
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Numerical dispersion
Solution
Provided that
Frequency fkc or cf/k is phase speed of
waves Waves with all wavelengths are propagated
with same speed gt no change in shape at a
constant velocity along x-axis
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Numerical dispersion
Hyrbid of differential and finite difference
equation semi-discrete equation
Solution
Provided that
Phase-speed dependent on k gt Dispersion of waves
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Numerical dispersion
Small klarge wavelength gt large wave
speed large k small wavelength gtwave speed goes
to zero
Check numerical scheme prior to any application
for numerical dispersion and diffusion!!!
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Summary of schemes
FTCS scheme
Unconditionally unstable diffusion coeff. Negative
Leapfrog scheme
Parasitic waves negative concentrations
UCTS scheme
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Summary of schemes
Matsuno scheme too diffusive no computational
mode
Lax-Wendroff scheme no computational mode,
damping of high freq, waves
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Summary of important concepts

• Galerkin approximation
• Von Neumann Stability criterion
• CFL criterion
• Numerical diffusion, bad for amplitude
• Numerical dispersion, bad for phase

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Term projects
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shallow-water model Thomas West-Antarctic
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Bifurcation analysis Laurie Ice-sheet model