Example - conservative Lorenz model

The famous Lorenz Attractor is obtained in the limit case of a coarse discretization of the Shallow Water Equations. Lorenz derived his model by truncating a spectral approximation of the SWEs on a double-periodic plane to just 3 modes. Since the SWEs have 3 variables, this leads to a system of 9 differential equations. Following the evolution of the vorticity coefficients in 3D space produces the attractor.

To illustrate the use of different discrete models, we consider the Lorenz system in the absence of damping and external forcing. This system was simplified by Lorenz to a 5-component model with the following properties:

Noncanonical Hamiltonian Structure - this implies conservation of total energy and a symplectic flow map
Enstrophy is conserved - this implies that the first three components evolve on a cylinder
Two-scale - the last two components are highly oscillatory
Chaotic - for some parameter ranges, there are sensitive initial conditions

We have simulated this problem using two numerical discretizations:

A Symplectic i.e. (Poisson) method (Movie: 6MB) based on a 3-term splitting. This method is only first order accurate, but conserves the Hamiltonian structure and enstrophy exactly. It also approximately conserves the energy and an adiabatic invariant associated with the fast components.
The 4th order Runge-Kutta method (Movie: 5MB). This method is highly accurate, but has none of the properties of the above method.

The figure to the right shows the phase space of the first three components as revealed by the two methods. The symplectic method does a good job of exploring the whole phase space as seen by the 'XX' pattern on the cylinder. On the other hand, the Runge-Kutta method spends most of its time in just one branch of phase space, and in fact settles into a artificial limit cycle.

We can also compare the methods based on the conserved quantities of the continuous system. For the Symplectic method, the enstrophy (red) is exactly conserved, the energy (green) is approximately conserved, and the oscillatory energy (blue) is also approximately conserved (this quantity, related to geostophic balance, is only approximately constant anyway). By comparison, for the Runge-Kutta method the energy decays by 80%, the oscillatory energy is completely damped out, and even the enstrophy is lost (the solution does not evolve on a cylinder).

For these examples, a very large stepsize is used to clearly see the qualitative differences between the two methods. For smaller time steps, the improved accuracy of the Runge-Kutta method will eventually make it more difficult to observe the effects seen here. However, if the integration is continued long enough, the same behavior will prevail: The very process of discretization with the Runge-Kutta method introduces a fundamentally different discrete dynamics, i.e. one with asymptotically stable limit cycles. Any statistical analysis based on ensemble simulations using the Runge-Kutta method will be skewed by the poor quality of this discrete model.

Return to HPM home page.