Improved Iterative Methods for the Solution Grid Equations: Theory and Application

Dear Colleagues,

This Special Issue aims to publish original research articles in constructing and investigating advanced difference schemes with improved dispersion and accuracy for diffusion–convection–reaction problems, which have arisen in modeling hydrophysical and hydrobiological processes for sea and coastal systems as well as for the Korteweg de Vries equation. The set of improved difference schemes, here presented and investigated, is based on linear combinations of leaf-frog and central difference schemes for relatively small grid Peclet numbers (less than number 2) and linear combinations of leaf-frog schemes and upwind schemes for large values of grid Peclet numbers. The original splitting schemes—two-dimensional–one-dimensional additive schemes—have been elaborated for convection–diffusion problems in natural systems. For the numerical solution of appropriate grid equations with non-self-adjoint operators, two variants of symmetric triangular–diagonal precondition methods have been built—one of variation type and the other using spectral estimations. The linearization on the time grid and convergence to the primary nonlinear task solutions of linearized boundary value problems has been investigated in L1 and L2 and its well-posedness. Additionally, investigations of related problems have been discussed—interpolation bottom boundary surfaces based on hyperbolic exponent splines. 

Potential topics include but are not limited to the following:

1. Construction and study of the leaf-frog ("cabaret") difference scheme with improved dispersion properties for the Korteweg de Vries equation;
2. Construction and study of the difference scheme leaf-frog ("cabaret") difference scheme with improved dispersion properties for the convection–diffusion equations;
3. Optimization of the schemes with weights for the numerical solution of the convection–diffusion equation;
4. Interpolation of reliefs and physical fields based on hyperbolic splines;
5. Construction and study of locally-two-dimensional–locally-one-dimensional schemes for convection–diffusion problems in natural systems;
6. Investigation of the convergence in L2 solutions of linearized on time grid chain boundary value problems for biogeochemical cycles to the origin nonlinear boundary value problem;
7. An improved iterative alternating–triangular method for solving the convection–diffusion grid equations with a bounded grid Peclet number based on a priori spectral estimates;
8. Adaptive iterative alternating–triangular method of variational type for solving the grid equations of convection–diffusion with a bounded grid Peclet number.

Prof. Dr. Alexey Beskopylny
Prof. Dr. Alexander Sukhinov
Guest Editors

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