NUMERICAL METHOD FOR MASS ADVECTION WONCHAI PROMNOPAS 13 FEB 2003

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NUMERICAL METHOD FOR MASS ADVECTION(WONCHAI
PROMNOPAS)13 FEB 2003
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Central difference
Forward difference
The value of the second derivative at P
Backward difference
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The Advection Equation
Numerical Solution of the Advection Equation
time derivative
n labels the time step and k labels the grid point
space derivative
The finite difference equation
Forward Time Centered Space (FTCS) method
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The Lax Method
This method is stable provided the time step
satisfies the Courant-Friedrichs-Lewy (CFL)
condition
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Clearly the wave is being artificially spread
out. Note however that there are no oscillations
and that the wave moves at the correct speed
(i.e. the peak is in the right place, although
significantly reduced in size).
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why there is a difference between the FTCS and
Lax methods, re-arrange the Lax method
This is the same as the FTCS method except for
the last term, which is a finite difference
approximation to a second order space derivative
of a. The Lax method applied to the advection
equation is equivalent to the FTCS method applied
to an advection-diffusion equation
diffusion coefficient
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The Lax-Wendroff Method
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We see that there remains some oscillation in the
trailing half of the wave and the peak height is
slightly reduced. Note also that the peak is not
quite in the correct position. Despite these
problems, the Lax-Wendroff method is far superior
to the FTCS and Lax methods.
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Courant-Friedrichs-Lewy(CFL or Courant) condition
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Physics , Science Chiang Mai University