Nonisospectral water wave field: Fast and adaptive modal identification and prediction via reduced-order nonlinear solutions

Real-world water wave fields exhibit significant nonlinear and nonisospectral characteristics, making it challenging to predict their evolution by relying solely on numerical simulation or exact solutions using integrable system theory. Hence, this paper introduces a fast and adaptive method of modal identification and prediction in nonisospectral water wave fields using the reduced-order nonlinear solution (RONS) scheme. Specifically, we discuss the coarse graining and mode extraction of wave field snapshots from the data-driven and physics-driven perspectives and utilize the RONS method for principle modal prediction of nonisospectral water wave fields. This is achieved by investigating the standard and nonisospectral Gardner system describing nonlinear water waves as a demonstration. Through detailed comparison and analysis, the fundamental solitary behaviors and dispersive effects in the Gardner system are discussed. Subsequently, a neighbor approximation is developed that combines the essences of symbolic precomputation and numerical computation in the RONS procedure, which exploits the locality of nonlinear interactions in water wave fields.

Figure 1

Eliminating redundant fast degrees of freedom in real random wave fields: (a) before filtering and (b) after filtering.

Figure 2

Comparison of dynamical predictions of solitary behavior under different approaches: (a) exact solution from integrable system theory; (b) RONS prediction with initial state ${\displaystyle A\left(0\right)=0.89,L\left(0\right)=1.8,x_c\left(0\right)=0}$; (c) DNSs with initial state of exact soliton envelope; (d) DNSs with initial state of reduced-order Gaussian ansatz.

Figure 3

Comparison of nonisospectral solitary dynamics: (a) nonisospectral RONS prediction, with $\epsilon =0.5$ and $\omega =1$, initial conditions $A\left(0\right)=0.89,L\left(0\right)=1.8$, and $x_c\left(0\right)=0$; (b) DNSs with $\varepsilon =0.5,\omega =1$; (c) nonisospectral RONS prediction with $\varepsilon =2,\omega =2$, and initial conditions $A\left(0\right)=0.89,L\left(0\right)=1.8,x_c\left(0\right)=0$; (d) DNSs with ${\displaystyle \varepsilon =2,\omega =2}$.

Figure 4

Prediction with POD bases: (a) four-mode degenerate RONS prediction of the isospectral case defined in Fig. 2; (b) nine-mode full POD reconstruction of the case defined in Fig. 2; (c) four-mode degenerate RONS prediction of the nonisospectral case defined in Figs. 3 and 3; (d) nine-mode degenerate RONS prediction with the full POD bases obtained from DNS data of isospectral evolution.

Figure 5

Characterization of the four-mode behavior of the Gardner system: (a) numerical reduced-order solutions (NRONS); (b) Fourier pseudospectral DNS.

Figure 6

Four-mode evolution: (a) NRONS; (b) MRONS with converting threshold of ${\displaystyle 2.5L}$; (c) absolute error.

Figure 7

Four-mode evolution: (a) NRONS; (b) MRONS with converting threshold of ${\displaystyle 2.0L}$; (c) absolute error.