International Conference «Mathematical and Informational Technologies, MIT-2013»
(X Conference «Computational and Informational Technologies for Science,
Engineering and Education»)

Vrnjacka Banja, Serbia, September, 5–8, 2013

Budva, Montenegro, September, 9-14, 2013

## On some problems of construction of difference schemes on moving grids

The approach to the monotonization of two-step explicit schemes is considered, based on using the scheme parameter, chosen by investigating the scheme differential approximations. The influence of constant, ``quasi-constant'' and variable parameter on the monotonicity of two-step explicit schemes is studied. For the constant scheme parameter the example is presented of the non-monotonicity preserving scheme with the absence of dispersion in the solution of the second differential approximation. The example of the scheme parameter is provided for the two-step explicit scheme on moving nonuniform grid to be monotonicity preserving. The connection of the consistent approximation of the Jacobian and the velocities of moving grid nodes with the geometric conservation law is shown. A new approach to the construction of divergent schemes on moving grids is suggested. By the example of the equidistribution method some peculiarities of the generation of grids adapting to discontinuous solutions are studied and the questions of the solvability of grid generation equations and the quality of grid adaptation are investigated. By the example of the predictor-corrector scheme with constant coefficients it is shown that TVD-schemes can increase a number of extremums. The implicit procedure for the smoothing of the control function is used, leading to better reproduction of discontinuous solutions.

Further on, the predictor-corrector scheme is generalized for the two-dimensional linear transport equation with variable coefficients. The method for approximation of the contravariant velocity components is described, which guarantees the fulfillment of the continuity equation for the grid functions of moving curvilinear grids. The method for determination of the scheme parameters is suggested, so that the monotonicity of the numerical solution is preserved. The fulfillment of the difference analogue of the geometric conservation law is proved, guaranteeing the preservation of the constant function by the predictor-corrector scheme. The modification of the classic equidistribution method is suggested, allowing to avoid the generation of oscillations of the node trajectories and the sharp changes of the areas of the adjacent grid cells. The method is given for the grid generation on the boundary of  a moving domain. The implicit procedure for smoothing of the control function is used.

 Abstracts file: Shokina_MIT-2013_abstract.doc Full text file: Shokina_MIT2013.pdf

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