Local two-dimensional splitting schemes for 3D suspended matter transport problem parallel solution

A 3D-model of suspended matter transport in coastal marine systems, which takes into account many factors, including the hydraulic size or the rate of particle deposition, the suspended matter propagation, sedimentation, the suspended matter sources distribution intensity, etc is considered. The difference operators of diffusion transport in the horizontal and vertical directions have significantly different characteristic spatiotemporal scales of processes, as well as spectra. With typical discretization, in relation to shallow-water systems in the Southern Russia (Azov Sea, Tsimlyansk reservoir), the steps in horizontal directions are 200–1000 meters, the turbulent exchange coefficients (turbulent diffusion) (10$^3$–10$^4$) m$^2$/sec; steps in vertical direction 0.1–1 m, and the vertical microturbulent exchange coefficients (0.1–1) m$^2$/sec. If focus on the using of explicit local 2D-1D splitting schemes, then the permissible values of the time step for 2D problem will be about 10–100 seconds, and for 1D problem in the vertical direction 0.1–1 sec. This motivates to construct an additive local 2D-1D splitting scheme in geometric directions. The paper describes parallel algorithm that uses both explicit and implicit schemes to approximate the 2D diffusion-convection problem in horizontal directions and 1D diffusion-convection problem in the vertical direction. The 2D implicit diffusion-convection problem in horizontal directions is numerically solved by the adaptive alternating-triangular method. The numerical implementation of 1D diffusion-convection problem in the vertical direction is carried out by sequential run-through method for series of independent 1D three-point problems in the vertical direction on given layer. To increase the efficiency of parallel calculations, the computational spatial grid and all grid data are also decomposed into one or two spatial directions in the horizontal directions. The obtained algorithms are compared taking into account the permissible time steps values and the actual time spent on performing calculations and exchanging information on each time layer.