```python
import numpy
import sympy
from matplotlib import pyplot
%matplotlib inline
```

```python
# Set the font family and size to use for Matplotlib figures.
pyplot.rcParams['font.family'] = 'serif'
pyplot.rcParams['font.size'] = 16
sympy.init_printing()
```

```python
# Set parameters.
nx = 101
L = 25.0
dx = L / (nx - 1)
dt = 0.001
nt = int(8/60/dt)
Vmax = 90.0
𝜌max = 100
x = numpy.linspace(0.0, L, num=nx)
```

```python
# u0 = numpy.ones(nx)
# for i in range(len(u0)):
#     if u0[i] == 1:
#         u0[i] = 10
# mask = numpy.where(numpy.logical_and(x >= 2.0, x <= 4.2))
# u0[mask] = 50.0
```

```python
def rho(x):
    rho = numpy.zeros_like(x)
    for i in range(len(rho)):
        if rho[i] == 0:
            rho[i] = 10
    mask = numpy.where(numpy.logical_and(x >= 2.0, x <= 4.2))
    rho[mask] = 50.0
    return rho
```

```python
rho0 = rho(x)
print(rho0)
```

    [10. 10. 10. 10. 10. 10. 10. 10. 50. 50. 50. 50. 50. 50. 50. 50. 50. 10.
     10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10.
     10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10.
     10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10.
     10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10.
     10. 10. 10. 10. 10. 10. 10. 10. 10. 10. 10.]

```python
def flux(rho, Vmax, 𝜌max):
    F = rho * Vmax * (1 - rho/𝜌max)
    return F
```

```python
def ftbs(rho0, nt, dt, dx, bc_value, *args):
    rho_hist = [rho0.copy()]
    rho = rho0.copy()
    for n in range(nt):
        F = flux(rho, *args)
        rho[1:] = rho[1:] - dt/dx * (F[1:] - F[:-1])
        rho[0] = bc_value
        rho_hist.append(rho.copy())
    return rho_hist
```

```python
rho_hist = ftbs(rho0, nt, dt, dx, rho0[0], Vmax, 𝜌max)
print(rho_hist)
```

    [array([10., 10., 10., 10., 10., 10., 10., 10., 50., 50., 50., 50., 50.,
           50., 50., 50., 50., 10., 10., 10., 10., 10., 10., 10., 10., 10.,
           10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10.,
           10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10.,
           10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10.,
           10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10.,
           10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10., 10.,
           10., 10., 10., 10., 10., 10., 10., 10., 10., 10.]), array([10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 44.24,
           50.  , 50.  , 50.  , 50.  , 50.  , 50.  , 50.  , 50.  , 15.76,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  , 10.  ,
           10.  , 10.  ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 38.59943936, 49.88056064,
           50.        , 50.        , 50.        , 50.        , 50.        ,
           50.        , 50.        , 19.98055936, 11.53944064, 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 33.30734138, 49.41270998,
           49.99994864, 50.        , 50.        , 50.        , 50.        ,
           50.        , 50.        , 23.2247599 , 13.62041276, 10.43482735,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 28.55046285, 48.41083019,
           49.99870697, 50.        , 50.        , 50.        , 50.        ,
           50.        , 50.        , 25.80564844, 15.80403194, 11.30577001,
           10.12454961, 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 24.44676036, 46.76362433,
           49.98961532, 49.99999999, 50.        , 50.        , 50.        ,
           50.        , 50.        , 27.91296837, 17.90642325, 12.48613514,
           10.4586588 , 10.03581444, 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 21.03744537, 44.45064618,
           49.95190885, 49.99999961, 50.        , 50.        , 50.        ,
           50.        , 50.        , 29.66918145, 19.85820177, 13.84438774,
           11.02107819, 10.15684091, 10.01030994, 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 18.29723182, 41.54172291,
           49.84105399, 49.99999128, 50.        , 50.        , 50.        ,
           50.        , 50.        , 31.15721331, 21.64087072, 15.27970881,
           11.78473916, 10.40207644, 10.05242269, 10.00296888, 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 16.15546766, 38.1810399 ,
           49.58359212, 49.99990033, 50.        , 50.        , 50.        ,
           50.        , 50.        , 32.4353955 , 23.25795329, 16.72423907,
           12.70240631, 10.78939824, 10.15255087, 10.01720172, 10.00085501,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 14.51909619, 34.56028751,
           49.0813402 , 49.99927611, 50.        , 50.        , 50.        ,
           50.        , 50.        , 33.54605069, 24.72179152, 18.1359402 ,
           13.72420954, 11.3162971 , 10.33380335, 10.05609956, 10.0055618 ,
           10.00024624, 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 13.29111651, 30.88645218,
           48.22619337, 49.99623794, 50.        , 50.        , 50.        ,
           50.        , 50.        , 34.52068751, 26.04751087, 19.49072994,
           14.80642234, 11.96608233, 10.61092518, 10.13568845, 10.02010545,
           10.00177701, 10.00007092, 10.        , 10.        , 10.        ,
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           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 11.23283148, 21.24108641,
           42.78136667, 49.86472502, 49.99999042, 50.        , 50.        ,
           50.        , 50.        , 36.84273511, 29.34231578, 23.12421136,
           18.11264416, 14.40826806, 12.02279616, 10.75231948, 10.21841001,
           10.04782039, 10.00759681, 10.00082597, 10.00005501, 10.00000169,
           10.        , 10.        , 10.        , 10.        , 10.        ,
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           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.88324756, 18.80814073,
           39.99148747, 49.6771997 , 49.99992454, 50.        , 50.        ,
           50.        , 50.        , 37.46594414, 30.25537045, 24.18825651,
           19.17282777, 15.30815257, 12.65458403, 11.10552417, 10.37031012,
           10.0967867 , 10.01917318, 10.00277577, 10.00027705, 10.00001705,
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           50.        , 50.        , 38.03151334, 31.09326268, 25.18328105,
           20.19547418, 16.2197076 , 13.34274496, 11.53068473, 10.57814557,
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           15.67034827, 14.96182183, 14.29294746, 13.66766659, 13.08979177,
           12.56277784, 12.08944876, 11.67170135, 11.31022181, 11.00426331,
           10.75153634, 10.54825402, 10.38934913, 10.2688435 , 10.18031448,
           10.11737961, 10.07411973, 10.04538167, 10.02693544, 10.01549554,
           10.00863988, 10.00466908, 10.00244566, 10.00124174, 10.00061117,
           10.00029163, 10.00013491, 10.00006051, 10.00002632, 10.0000111 ,
           10.00000454, 10.0000018 , 10.00000069, 10.00000026, 10.00000009,
           10.00000003, 10.00000001, 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.00000001, 10.00000005, 10.00000043, 10.00000321, 10.00002231,
           10.00014172, 10.00081711, 10.00426368, 10.02015505, 10.08653682,
           10.33767511, 11.18693159, 13.62910416, 18.89964537, 25.87850758,
           30.62062208, 32.04617898, 31.80864171, 31.03443765, 30.11096281,
           29.14968345, 28.18230497, 27.21838087, 26.26125106, 25.31252073,
           24.3733671 , 23.44494296, 22.52851538, 21.62552526, 20.73762318,
           19.86669796, 19.01490219, 18.18467417, 17.37875386, 16.60018842,
           15.85232137, 15.13875802, 14.46329827, 13.82982805, 13.24216185,
           12.70383351, 12.21784022, 11.78635625, 11.41044746, 11.08983006,
           10.82272425, 10.60584812, 10.43457738, 10.30326358, 10.20566769,
           10.1354374 , 10.08654875, 10.05364587, 10.03224315, 10.01878848,
           10.01061372, 10.00581248, 10.00308595, 10.00158845, 10.00079277,
           10.00038365, 10.00018004, 10.00008194, 10.00003616, 10.00001548,
           10.00000643, 10.00000259, 10.00000101, 10.00000038, 10.00000014,
           10.00000005, 10.00000002, 10.00000001, 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.00000004, 10.00000032, 10.00000241, 10.00001681,
           10.00010733, 10.0006226 , 10.00327113, 10.01557973, 10.06744437,
           10.26573081, 10.94700694, 12.96810018, 17.61944935, 24.4911192 ,
           29.87799209, 31.85458482, 31.83955069, 31.13799899, 30.24013529,
           29.29066629, 28.33089899, 27.37314614, 26.42154477, 25.47791708,
           24.54347675, 23.61935123, 22.70675647, 21.80706732, 20.92185635,
           20.05292306, 19.20231961, 18.37237339, 17.56570454, 16.78523459,
           16.03418092, 15.31603007, 14.63448179, 13.99335506, 13.39644822,
           12.84734884, 12.34919544, 11.90440374, 11.51438305, 11.179282  ,
           10.89781165, 10.66719308, 10.48326185, 10.34073316, 10.2335965 ,
           10.15557749, 10.10058961, 10.06310529, 10.03840052, 10.02266096,
           10.01296718, 10.00719495, 10.0038711 , 10.00201971, 10.00102192,
           10.00050148, 10.00023868, 10.00011019, 10.00004935, 10.00002144,
           10.00000903, 10.00000369, 10.00000146, 10.00000056, 10.00000021,
           10.00000008, 10.00000003, 10.00000001, 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.00000003, 10.00000024, 10.00000181, 10.00001266,
           10.00008126, 10.0004742 , 10.00250839, 10.01203569, 10.05252286,
           10.20886215, 10.75377377, 12.4145114 , 16.45714779, 23.05904978,
           28.99308402, 31.58228402, 31.84151567, 31.23150238, 30.36497311,
           29.42899359, 28.47732336, 27.52587462, 26.5798333 , 25.64131748,
           24.71160404, 23.79180582, 22.88309352, 21.98678044, 21.10436549,
           20.23756339, 19.38833053, 18.55888812, 17.75174086, 16.96968822,
           16.21582333, 15.49351332, 14.80635338, 14.15808603, 13.55247771,
           12.99314699, 12.48334439, 12.02569268, 11.62190878, 11.27254156,
           10.9767704 , 10.73231192, 10.53547227, 10.38135863, 10.26423034,
           10.17793766, 10.11637541, 10.07387869, 10.04550646, 10.02719049,
           10.01575775, 10.00885694, 10.00482824, 10.00255287, 10.00130927,
           10.00065136, 10.00031437, 10.0001472 , 10.00006687, 10.00002947,
           10.00001261, 10.00000523, 10.00000211, 10.00000082, 10.00000031,
           10.00000011, 10.00000004, 10.00000001, 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.00000002, 10.00000018, 10.00000136, 10.00000954,
           10.0000615 , 10.00036104, 10.00192257, 10.00929233, 10.04087196,
           10.16398354, 10.59872761, 11.95516104, 15.42198212, 21.62154049,
           27.96879661, 31.2148023 , 31.80738152, 31.3125958 , 30.48477311,
           29.56446888, 28.62153685, 27.67657314, 26.73613773, 25.80274582,
           24.87777124, 23.96232512, 23.05753935, 22.16467031, 21.2851475 ,
           20.42060517, 19.57290856, 18.74417727, 17.93680499, 17.15347282,
           16.39715208, 15.67109063, 14.97877588, 14.32386615, 13.71008241,
           13.14105403, 12.62011659, 12.15006752, 11.7328963 , 11.3695189 ,
           11.05955748, 10.80121188, 10.59126371, 10.42523469, 10.29769107,
           10.20265259, 10.13404011, 10.08608864, 10.05366547, 10.0324607 ,
           10.01904861, 10.01084376, 10.0059883 , 10.00320812, 10.00166741,
           10.00084084, 10.00041142, 10.00019534, 10.00009   , 10.00004024,
           10.00001746, 10.00000736, 10.00000301, 10.00000119, 10.00000046,
           10.00000017, 10.00000006, 10.00000002, 10.00000001, 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ]), array([10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.00000002, 10.00000013, 10.00000102, 10.00000718,
           10.00004654, 10.00027477, 10.00147286, 10.00717013, 10.03178273,
           10.1286182 , 10.47471502, 11.57697926, 14.5156085 , 20.21645201,
           26.81692967, 30.73783732, 31.72849729, 31.37828755, 30.59862313,
           29.69682954, 28.76347677, 27.82524126, 26.89047657, 25.96222518,
           25.04200061, 24.13092801, 23.23010777, 22.34074412, 21.46420133,
           20.60203741, 19.75603082, 18.92820409, 18.12084426, 17.33651797,
           16.57807742, 15.8486521 , 15.15161952, 14.49054755, 13.86910014,
           13.29089963, 12.75934222, 12.27736966, 11.84721012, 11.47011314,
           11.14611586, 10.87388482, 10.65067633, 10.47244348, 10.33409169,
           10.22985248, 10.15371734, 10.09986066, 10.06298703, 10.0385611 ,
           10.0229088 , 10.01320587, 10.00738638, 10.00400872, 10.00211111,
           10.00107888, 10.00053509, 10.00025757, 10.00012034, 10.00005458,
           10.00002402, 10.00001027, 10.00000426, 10.00000171, 10.00000067,
           10.00000025, 10.00000009, 10.00000003, 10.00000001, 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        , 10.        , 10.        , 10.        , 10.        ,
           10.        ])]

```python
def ftbs1(rho0, nt, dt, dx, *args):
    rho = rho0.copy()
    for n in range(1, nt):
        F = flux(rho, *args)
        rho[1:] = rho[1:] - dt/dx * (F[1:] - F[:-1])
        rho[0] = bc_value
    return rho

rho = ftbs1(rho0, nt, dt, dx, rho0[0], Vmax, 𝜌max)
print(rho)
```

```python

```