Compressive spectral method for the simulation of the water waves

In this paper an approach for decreasing the computational effort required for the spectral simulations of the water waves is introduced. Signals with majority of the components zero, are known as the sparse signals. Like majority of the signals in the nature it can be realized that water waves are sparse either in time or in the frequency domain. Using the sparsity property of the water waves in the time or in the frequency domain, the compressive sampling algorithm can be used as a tool for improving the performance of the spectral simulation of the water waves. The methodology offered in this paper depends on the idea of using a smaller number of spectral components compared to the classical spectral method with a high number of components. After performing the time integration with a smaller number of spectral components and using the compressive sampling technique, it is shown that the water wave field can be reconstructed with a significantly better efficiency compared to the classical spectral method with a high number of spectral components, especially for long time evolutions. For the sparse water wave model in the time domain the well-known solitary wave solutions of the Korteweg-deVries (KdV) equation is considered. For the sparse water wave model in the frequency domain the well-known Airy (linear) ocean waves with Jonswap spectrum is considered. Utilizing a spectral method, it is shown that by using a smaller number of spectral components compared to the classical spectral method with a high number of components, it is possible to simulate the sparse water waves with negligible error in accuracy and a great efficiency especially for large time evolutions.