Parallelization of Numerical Algorithms for Solving Mathematical Models of Hereditary Dynamical Systems in Geospheres. Solution of Test Problems on GPU Computing Cluster and Efficiency Analysis

The article considers program implementations of numerical algorithms for solving model equations taking into account the memory effect, adapted for execution on a computer system with parallel architecture, and also presents an analysis of parallelization efficiency. Memory (or legacy properties) are observed in many dynamic processes and mathematical models, and can be described using fractional derivatives of constant or variable order. The memory effect is non-locality, i.e. dependence of the current value of the solution on previous values. This means that the numerical solution of such model equations can create a high load on computational units, which will seriously slow down computations. The article considers several test cases based on mathematical models of heredity dynamical systems in geospheres solved by a non-local implicit finite difference scheme. By analyzing data on average computation time for different implementations of algorithms on CPU and/or GPU computational units of a GPU cluster, it is shown that it is possible to achieve a reduction of computation time up to 12–14 times, but there are limitations on the number of parallel threads up to which parallel implementation is effective.