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PDEs Numerical Methods Course

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    0 1 2 3 4 5 6

    1

    0

    1

    mesh[sin(x

    2)]

    x

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    0 1 2 3 4 5 61

    0

    1

    LFEM

    [sin(x

    2)]

    0 1 2 3 4 5 6

    0

    1

    2

    e5 > f

    5=0.947

    x

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    0 1 2 3 4 5 61

    0

    1

    spline[sin(x

    2)]

    x

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    0 2 4 6 8 10 12

    1

    0

    1

    FT3

    0 2 4 6 8 10 12

    1

    0

    1

    FFT[square2

    (x)]

    FT6

    0 2 4 6 8 10 12

    1

    0

    1

    FT12

    x

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    0 1 2 3 4 5 61

    0

    1V5

    0 1 2 3 4 5 61

    0

    1 V4

    Haarwavelets

    [sin(x

    2)]

    0 1 2 3 4 5 61

    0

    1W4

    x

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    0 1 2 3 4 5 6

    1

    0

    1 V5

    0 1 2 3 4 5 6

    1

    0

    1 V4

    DAUB4wavelets[sin(x

    2)]

    0 1 2 3 4 5 6

    1

    0

    1 W4

    x

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    10 5 0 5 100

    1000

    2000

    3000

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    t + t

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    t

    x

    t

    j1 j j+1

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    1

    1 Im G

    Re G

    1

    exp( )

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    t + t

    t t

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    1 1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    1 1 1 1

    x

    t

    j1 j j+1

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    j+1j1 j0 0 0

    0 0 0

    0 0 0

    0 0 0

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    1 1 1 1

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    1 1 1

    1 1 1

    1 1 1

    1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    1 1 1 1

    t+ t

    x

    t

    t

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    t + t

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    0 0 0 0

    0 0 0 0

    0 0 0 0

    1 1 1 1

    1 1 1 1

    1 1 1 1

    t

    x

    t

    j1 j j+1

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    0

    aliasing

    aliased spectrumtrue spectrum

    kkc c

    A(k)

    k

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