Koopman analysis of Burgers equation

The emergence of dynamic mode decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear partial differential equation (PDE) presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis, and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in the Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions. As far as we are aware the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them, and it presents a nice example in which (i) the Koopman modes are linearly dependent and so they cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions, and (ii) the Koopman eigenvalues are highly degenerate, which means that computed Koopman modes become initial-condition-dependent. As a way of illustration, we discuss the form of the Koopman expansion with various initial conditions, and we assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.

Figure 1

Evolution of (left) $u_G\left(x\right)=\text{exp}[-{(x+\pi /2)}^2]$ and (right) $u_S\left(x\right)=\text{sin}(2\pi x/L)$ for $t\le 40$ [blue (red) profile is the initial (final) state].

Figure 2

(Top) Koopman eigenvalues for the Burgers equation (filled circles) with the eigenvalues required to evolve $v\left(u\right)$ overlaid as crosses. (Bottom) The first eight nontrivial Koopman modes, summed over their respective degeneracies, evaluated with $u=u_G$ (blue) and $u=u_G(x-\pi /2)$ (dashed red): The shapes of Koopman modes 4 and above are dependent on initial conditions whereas those of modes 1–3 are not.

Figure 3

Left: Eigenvalues from DMD of $u_G$ with $M=200$ snapshot pairs with spacing $\delta t=1$, collected within the time window $t\in [0,80)$ for both $\mathbf{\psi }=\mathbf{u}$ (red) and $\mathbf{\psi }=\mathbf{v}\left(\mathbf{u}\right)$ (blue). Note the crosses are the eigenvalues of the linear diffusion problem ${\lambda }_n=-n^2{\pi }^2\nu /L^2$. The rank of the problem $r=70$ for $\mathbf{\psi }=\mathbf{u}$ and $r=45$ for $\mathbf{\psi }=\mathbf{v}\left(\mathbf{u}\right)$. Center: The error in the approximation ${\mathbf{\psi }}^{-1}\left({\sum }_j{a}_j{\widehat{\mathbf{\psi }}}_j^D\text{exp}\left({\lambda }_{j}t\right)\right)$ as compared with the state vector $\mathbf{u}$, evaluated beyond the observation window (shown in gray; colors as in the left panel). The $a_j$ are determined from a least-squares fit for the observable over the DMD window. Right: Koopman eigenfunctions $\left\{{\phi }_n\left(u\right)\right\}$ ($n=1,\cdots ,6$) required to evolve $v\left(u\right)$: blue lines are a direct evaluation of ${\phi }_n\left(u\right)=\langle {\widehat{v}}_n,v\left(u\right)\rangle$ on the trajectory $f^t\left(u_G\right)$; dashed, black lines are Koopman eigenfunctions obtained from the DMD with $\mathbf{\psi }=\mathbf{v}\left(\mathbf{u}\right)$.