Absolute stability of explicit difference schemes for the heat equation under Fourier–Tikhonov regularization

The paper considers the possibility of constructing a simple and absolutely stable explicit difference scheme for the heat equation. Due to the too rigid stability condition, the invention of the sweep method for solving SLAE with three diagonal matrices, and splitting schemes, absolutely stable implicit schemes forced out explicit schemes for the heat equation of programming practice. However, implicit schemes are poorly parallelized. Therefore, programs for solving problems of heat conduction, diffusion, underground hydrodynamics, etc. on huge spatial grids using multiprocessor computing systems require using explicit difference schemes. This is especially true for multiprocessor systems of teraflop and higher performance that combine hundreds of processors. In this case, explicit schemes must be absolutely stable or, in the extreme case, their stability condition must be no more stringent than the same for hyperbolic equations. The paper proposes modifications of explicit difference schemes that approximate a parabolic equation and have absolute countable stability. Countable stability of a solution obtained at each time step by the classical explicit scheme is achieved by the fast Fourier transform and the subsequent Fourier synthesis with Tikhonov regularization. When calculating the direct and inverse Fourier transforms, the author used the Cooley–Tukey algorithm of the fast Fourier transform. There are the results of comparing numerical calculations of model problems with analytical solutions. The absolute stability of the proposed explicit schemes for the heat equation allows their wide use for parallel computations.