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in Burgers, the import statements appeared twice #221

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in Burgers, the import statements appeared twice #221

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53 changes: 11 additions & 42 deletions Burgers.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -7,38 +7,6 @@
"# Burgers' equation"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": [
"hide"
]
},
"outputs": [],
"source": [
"%matplotlib inline"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"tags": [
"hide"
]
},
"outputs": [],
"source": [
"%config InlineBackend.figure_format = 'svg'\n",
"from ipywidgets import interact\n",
"from ipywidgets import widgets\n",
"from ipywidgets import FloatSlider, fixed\n",
"from exact_solvers import burgers\n",
"from exact_solvers import burgers_demos\n",
"from IPython.display import HTML"
]
},
{
"cell_type": "markdown",
"metadata": {},
Expand Down Expand Up @@ -81,15 +49,6 @@
"However, since the charactistic speed depends on the solution, these lines are not parallel and characterstics may converge or spread out."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Shock formation\n",
"\n",
"In the figure below we consider Burgers' equation with a Gaussian hump as the initial data. Since the characteristic speed in Burgers' equation is given by $q$ itself, the peak of the hump travels faster than the rest, and characteristics are converging at the front of the traveling wave (where $f'(q)$ decreases with $x$) while they are spreading out behind the peak (where $f'(q)$ increases with $x$). The dashed line shows the initial condition while the solid lines show the solution at later times."
]
},
{
"cell_type": "code",
"execution_count": null,
Expand Down Expand Up @@ -118,7 +77,17 @@
"from ipywidgets import widgets\n",
"from ipywidgets import FloatSlider, fixed\n",
"from exact_solvers import burgers\n",
"from exact_solvers import burgers_demos"
"from exact_solvers import burgers_demos\n",
"from IPython.display import HTML"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Shock formation\n",
"\n",
"In the figure below we consider Burgers' equation with a Gaussian hump as the initial data. Since the characteristic speed in Burgers' equation is given by $q$ itself, the peak of the hump travels faster than the rest, and characteristics are converging at the front of the traveling wave (where $f'(q)$ decreases with $x$) while they are spreading out behind the peak (where $f'(q)$ increases with $x$). The dashed line shows the initial condition while the solid lines show the solution at later times."
]
},
{
Expand Down