Safe zone for phaseresolved simulation of interactions between waves and vertically sheared currents Abstract   In this letter, we analyze the numerical stability of a velocitybased boundary integral equation for nonlinear interactions between surface waves and vertically sheared currents and propose an upper bound for the wave elevations associated with a finite number of resolved wave modes to guarantee the convergence of the numerical solution. The upper bound is expressed as a function of the      L    xe2x88x9e      norm of dimensionless wave elevations and wave steepness. In general, energy accumulation at high wavenumbers may lead to numerical stability. The upper bound decreases as we increase the number of resolved wave modes. Furthermore, when we assume the regularity of the wave field, such as the powerlaw decay in the spectral domain, the upper bound become independent of the number of resolved wave modes when it becomes large enough, and the criteria of simulation stability are relaxed.

Stable and Accurate Filtering Procedures High frequency errors are always present in numerical simulations since no difference stencil is accurate in the vicinity of the pi mode. To remove the defective high wave number information from the solution, artificial dissipation operators or filter operators may be applied. Since stability is our main concern, we are interested in schemes on summationbyparts (SBP) form with weak imposition of boundary conditions. Artificial dissipation operators preserving the accuracy and energy stability of SBP schemes are available. However, for filtering procedures it was recently shown that stability problems may occur, even for originally energy stable (in the absence of filtering) SBP based schemes. More precisely, it was shown that even the sharpest possible energy bound becomes very weak as the number of filtrations grow. This suggest that successful filtering include a delicate balance between the need to remove high frequency oscillations (filter often) and the need to avoid possible growth (filter seldom). We will discuss this problem and propose a remedy.