Economical Explicit-Implicit Schemes for Solving Multidimensional Diffusion–Convection Problems

Abstract

The goal of this work is to construct an efficient parallel numerical solution for a nonstationary diffusion–convection problem on a multiprocessor computer system with distributed memory. Economical explicit-implicit difference schemes and the method of splitting into physical processes are taken as a basis. When using these schemes, it becomes possible to pass to a chain of one-dimensional and two-dimensional difference problems that approximate the original problem in an overall sense. The explicit-implicit difference schemes suggest an explicit approximation in horizontal directions and an implicit approximation with weights in a vertical direction, taking less time to solve the diffusion–convection problem compared to the explicit schemes while retaining an admissible accuracy of the solution. We propose an algorithm for finding an optimal weight that provides a minimum approximation error in the solution of the diffusion–convection problem in a vertical direction for given time grid steps. Computational experiments have been carried out using the three-dimensional model problem of suspension transport in a water environment as an example. The model takes into account the following processes: the advective transport due to the motion of the water environment, the microturbulent diffusion and gravitational settling of suspension particles, and the change in the geometry of the bottom caused by the settling of suspension particles or the rise of bottom sediment particles. The presented parallel approach to the numerical simulation of suspension transport processes will allow one to improve the real-time forecast accuracy manifold and the validity of the engineering decisions being made when creating the objects of coastal infrastructure.