We extend the Virtual Element Method to a two-dimensional unsteady nonlinear convection-diffusion equation characterized by a fractional-order derivative with respect to the time variable. Our methodology is based on three fundamental technical components: a fractional version of the Grunwald-Letnikov approximation, discrete maximal regularity, and the regularity theory associated with non-linearity. We prove the method's well-posedness, i.e., the approximate solution's existence and uniqueness to the time-fractional convection-diffusion equation with a Lipschitz nonlinear source term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the $L^2$-norm and $H^1$-norm to various mesh configurations is validated by numerical results, underlining the practical effectiveness of the proposed method.

In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.

This study examines the dynamic response of a fibrillar adhesive floating breakwater positioned near a porous structure at a finite distance from Gaussian undulating seabed. The problem is addressed using linearized water wave theory, with numerical simulations based on the multi-domain boundary element method. The study primarily focuses on the analysis of crucial elements such as the added mass and damping coefficients associated with heave, surge, and pitch motions, considering the influence of both wave and structural parameters. Validation against existing literature confirms the accuracy and reliability of the proposed method. The study reveals that an increase in the number of seabed ripples leads to higher added mass and damping coefficients, particularly at larger wave incidence angles. Further, the frictional interaction between the water and the porous structure modifies the added mass coefficient, resulting in a shift in the resonance peak and significantly affecting the dynamic response of the breakwater. Moreover, surge and pitch motions are highly damped in intermediate waves as the porosity of the structure decreases.