Economic explicit-implicit schemes for solving multidimensional diffusion-convection problems

Abstract

This work concerns the development of an efficient parallel algorithm for numerical solution of nonstationary diffusion-convection problems by means of a multiprocessor computer system with distributed memory. Economically explicit-implicit difference schemes and the method of splitting into physical processes are used as a basis. The original problem is replaced by a sequence of one-dimensional and two-dimensional difference problems using complex schemes that approximate the original problem in the general sense. Explicit-implicit difference schemes involve explicit approximation in horizontal directions and implicit approximation with weights in a vertical direction and require less time for solving diffusion-convection problems compared to explicit schemes while maintaining acceptable accuracy of solutions. The algorithm is proposed to find the optimal weight value and it yields the lowest approximation error in the solution of the diffusion-convection problem in the vertical direction for given time grid steps. The three-dimensional model problem of transport of suspensions in the water environment is considered. The model takes into account the following processes: advective transport caused by the movement of the water medium, microturbulent diffusion and gravitational deposition of suspended particles, as well as changes in the geometry of the bottom caused by the deposition of suspended particles or the rise of sediment particles. Application of the parallel algorithm developed for numerical modeling of suspension transport makes it possible to significantly improve the real-time forecast accuracy and the validity of the accepted engineering decisions used in designing the objects of coastal infrastructure.