implementations.problem_classes.FermiPastaUlamTsingou module¶
- class fermi_pasta_ulam_tsingou(npart=2048, alpha=0.25, k=1.0, energy_modes=None)[source]¶
Bases:
ptypeThe Fermi-Pasta-Ulam-Tsingou (FPUT) problem was one of the first computer experiments. E. Fermi, J. Pasta and S. Ulam investigated the behavior of a vibrating spring with a weak correction term (which is quadratic for the FPU-\(\alpha\) model, and cubic for the FPU-\(\beta\) model [1]). This can be modelled by the second-order problem
\[\frac{d^2 u_j(t)}{d t^2} = (u_{j+1}(t) - 2 u_j(t) + u_{j-1}(t)) (1 + \alpha (u_{j+1}(t) - u_{j-1}(t))),\]where \(u_j(t)\) is the position of the \(j\)-th particle. [2] is used as setup for this implemented problem class. The Hamiltonian of this problem (needed for second-order SDC) is
\[\sum_{i=1}^n \frac{1}{2}v^2_{i-1}(t) + \frac{1}{2}(u_{i+1}(t) - u_{i-1}(t))^2 + \frac{\alpha}{3}(u_{i+1}(t) - u_{i-1}(t))^3,\]where \(v_j(t)\) is the velocity of the \(j\)-th particle.
- Parameters:
npart (int, optional) – Number of particles.
alpha (float, optional) – Factor of the nonlinear force \(\alpha\).
k (float, optional) – Mode for initial conditions.
energy_modes (list, optional) – Energy modes.
- dx¶
Mesh grid size.
- Type:
float
- xvalues¶
Spatial grid.
- Type:
np.1darray
- ones¶
Vector containing ones.
- Type:
np.1darray
References
- dtype_f¶
alias of
acceleration
- dtype_u¶
alias of
particles
- eval_f(u, t)[source]¶
Routine to compute the right-hand side of the problem.
- Parameters:
u (dtype_u) – Current values of the numerical solution.
t (float) – Current time of the numerical solution is computed.
- Returns:
f – The right-hand side of the problem.
- Return type:
dtype_f
- eval_hamiltonian(u)[source]¶
Routine to compute the Hamiltonian.
- Parameters:
u (dtype_u) – The particles.
- Returns:
ham – The Hamiltonian.
- Return type:
float