Improved Iterative Methods for the Solution Grid Equations: Theory and Application, 2nd Edition

Dear Colleagues,

The aim of this Special Issue is to publish original research articles regarding the construction and investigation of advanced difference schemes with improved dispersion and accuracy for diffusion–convection–reaction problems, which have arisen when 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 presented and investigated here is based on linear combinations of leaf-frog and central difference schemes for relatively small grid Peclet numbers (less than 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 upon for use in solving 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 the variation type and the other using spectral estimations. 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, as well as their 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 convergence in L2 solutions of problems regarding linearization on time grid chain boundary values for biogeochemical cycles to the problem of the origin of nonlinear boundary values;
7. An improved iterative alternating–triangular method to solve 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 to solve the grid equations of convection–diffusion with a bounded grid Peclet number.

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

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• diffusion–convection problems
• difference schemes
• splitting schemes
• dispersion
• accuracy
• boundary value problems
• quasi-linear parabolic equations
• linearization
• convergence in spaces L1, L2
• grid equations
• non-self-adjoint operators
• grid Péclet number
• iterative symmetric triangular methods