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SYLLABUS
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2.5.2 Schrödinger equation
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2 FINITE DIFFERENCES
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2.7 Exercises
2.6 Computer quiz
A Courant-Friedrichs-Levy (CFL) number larger than unity (
) means that
the advection velocity exceeds the characteristic speed of the mesh
explicit and implicit finite difference schemes are always unstable
the numerical solution obtained are necessarily inaccurate
Dividing the time step by two to converge an explicit scheme for diffusion, you
multiply the mesh interval size by two
keep the mesh interval size fixed
divide the mesh interval size by two
divide the mesh interval size by four
A numerical scheme is necessarily unstable if the amplification factor
has a real part exceeding unity for one particular wavelength
has an imaginary part exceeding unity for all the wavelengths
has a norm exceeding unity for one particular wavelength
has a real part smaller than zero for short wavelengths
Implicit schemes such as Crank-Nicholson
are always slower than explicit schemes for the same degree of precision
require the more arithmetic operations per step than explicit schemes in 1D
require the more arithmetic operations per step than explicit schemes in 2D
cannot be used for 2D problems
An initial box function
is often used when a physical solution is sought for advection problems
should always be avoided to minimize inaccuracies in the physical solution
can make explicit finite difference schemes unstable
Compared with other methods, explicit finite difference schemes
are easy and robust to execute
are easy to understand and implement in a code
converge rather slowly
should be avoided for wave equations
SYLLABUS
Previous:
2.5.2 Schrödinger equation
Up:
2 FINITE DIFFERENCES
Next:
2.7 Exercises
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Lifelong-learners
at 04:57:52, March 18th, 2018