# Camassa-Holm equation¶

This tutorial was contributed by Colin Cotter.

The Camassa Holm equation [CH93] is an integrable 1+1 PDE which may be written in the form

solved in the interval \([a,b]\) either with periodic boundary conditions or with boundary conditions u(a)=u(b)=0; \(\alpha>0\) is a constant that sets a lengthscale for the solution. The solution is entirely composed of peaked solitons corresponding to Dirac delta functions in \(m\). Further, the solution has a conserved energy, given by

In this example we will concentrate on the periodic boundary conditions case.

A weak form of these equations is given by

Energy conservation then follows from substituting the second equation into the first, and then setting \(p=u\),

If we choose the same continuous finite element spaces for \(m\) and \(u\) then this proof immediately extends to the spatial discretisation, as noted by [Mat10]. Further, it is a property of the implicit midpoint rule time discretisation that any quadratic conserved quantities of an ODE are also conserved by the time discretisation (see [Ise09], for example). Hence, the fully discrete scheme,

where \(u^{n+1/2}=(u^{n+1}+u^n)/2\), \(m^{n+1/2}=(m^{n+1}+m^n)/2\), conserves the energy exactly. This is a useful property since the energy is the square of the \(H^1\) norm, which guarantees regularity of the numerical solution.

As usual, to implement this problem, we start by importing the Firedrake namespace.

```
from firedrake import *
```

To visualise the output, we also need to import matplotlib.pyplot to display the visual output

```
try:
import matplotlib.pyplot as plt
except:
warning("Matplotlib not imported")
```

We then set the parameters for the scheme.

```
alpha = 1.0
alphasq = Constant(alpha**2)
dt = 0.1
Dt = Constant(dt)
```

These are set with type `Constant`

so that the values can be
changed without needing to regenerate code.

We use a `periodic mesh`

of width 40
with 100 cells,

```
n = 100
mesh = PeriodicIntervalMesh(n, 40.0)
```

and build a `mixed function space`

for the
two variables.

```
V = FunctionSpace(mesh, "CG", 1)
W = MixedFunctionSpace((V, V))
```

We construct a `Function`

to store the two variables at time
level `n`

, and `split()`

it so that we can
interpolate the initial condition into the two components.

```
w0 = Function(W)
m0, u0 = w0.split()
```

Then we interpolate the initial condition,

into u,

```
x, = SpatialCoordinate(mesh)
u0.interpolate(0.2*2/(exp(x-403./15.) + exp(-x+403./15.))
+ 0.5*2/(exp(x-203./15.)+exp(-x+203./15.)))
```

before solving for the initial condition for `m`

. This is done by
setting up the linear problem and solving it (here we use a direct
solver since the problem is one dimensional).

```
p = TestFunction(V)
m = TrialFunction(V)
am = p*m*dx
Lm = (p*u0 + alphasq*p.dx(0)*u0.dx(0))*dx
solve(am == Lm, m0, solver_parameters={
'ksp_type': 'preonly',
'pc_type': 'lu'
}
)
```

Next we build the weak form of the timestepping algorithm. This is expressed
as a mixed nonlinear problem, which must be written as a bilinear form
that is a function of the output `Function`

`w1`

.

```
p, q = TestFunctions(W)
w1 = Function(W)
w1.assign(w0)
m1, u1 = split(w1)
m0, u0 = split(w0)
```

Note the use of `split(w1)`

here, which splits up a
`Function`

so that it may be inserted into a UFL
expression.

```
mh = 0.5*(m1 + m0)
uh = 0.5*(u1 + u0)
L = (
(q*u1 + alphasq*q.dx(0)*u1.dx(0) - q*m1)*dx +
(p*(m1-m0) + Dt*(p*uh.dx(0)*mh -p.dx(0)*uh*mh))*dx
)
```

Since we are in one dimension, we use a direct solver for the linear system within the Newton algorithm. To do this, we assemble a monolithic rather than blocked system.

```
uprob = NonlinearVariationalProblem(L, w1)
usolver = NonlinearVariationalSolver(uprob, solver_parameters=
{'mat_type': 'aij',
'ksp_type': 'preonly',
'pc_type': 'lu'})
```

Next we use the other form of `split()`

, `w0.split()`

,
which is the way to split up a Function in order to access its data
e.g. for output.

```
m0, u0 = w0.split()
m1, u1 = w1.split()
```

We choose a final time, and initialise a `File`

object for
storing `u`

. as well as an array for storing the function to be visualised:

```
T = 100.0
ufile = File('u.pvd')
t = 0.0
ufile.write(u1, time=t)
all_us = []
```

We also initialise a dump counter so we only dump every 10 timesteps.

```
ndump = 10
dumpn = 0
```

Now we enter the timeloop.

```
while (t < T - 0.5*dt):
t += dt
```

The energy can be computed and checked.

```
#
E = assemble((u0*u0 + alphasq*u0.dx(0)*u0.dx(0))*dx)
print("t = ", t, "E = ", E)
```

To implement the timestepping algorithm, we just call the solver, and assign
`w1`

to `w0`

.

```
#
usolver.solve()
w0.assign(w1)
```

Finally, we check if it is time to dump the data. The function will be appended to the array of functions to be plotted later:

```
#
dumpn += 1
if dumpn == ndump:
dumpn -= ndump
ufile.write(u1, time=t)
all_us.append(Function(u1))
```

This solution leads to emergent peakons (peaked solitons); the left peakon is travelling faster than the right peakon, so they collide and momentum is transferred to the right peakon.

At last, we call the function `plot`

on the final
value to visualize it:

```
try:
fig, axes = plt.subplots()
plot(all_us[-1], axes=axes)
except Exception as e:
warning("Cannot plot figure. Error msg: '%s'" % e)
```

And finally show the figure:

```
try:
plt.show()
except Exception as e:
warning("Cannot show figure. Error msg: '%s'" % e)
```

Images of the solution at shown below.

A python script version of this demo can be found here.

References

Roberto Camassa and Darryl D. Holm. An integrable shallow water equation with peaked solitons. *Physical Review Letters*, 71(11):1661, 1993.

Arieh Iserles. *A first course in the numerical analysis of differential equations*. Number 44 in Cambridge Texts in Applied Mathematics. Cambridge University Press, 2009.

Takayasu Matuso. A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation. *Journal of Computational and Applied Mathematics*, 234(4):1258–1266, 2010.