Some aspects of compact difference scheme convergence

Difference schemes, compact on space variables, i.e. constructed for each space direction on the two-or three-dot stencil, have the advantages of efficiency and convenience of boundary conditions formulation in comparison with other schemes of the high order of accuracy. Originally these schemes were developed for receiving smooth solutions. In the last two decades compact schemes are actively used for calculating gas dynamics flows with shock waves. However to obtain numerical solution with the guaranteed accuracy the knowledge of real properties of difference schemes at calculation of solutions with features (breaks) is required. Now this question for of some widely used compact schemes is no yet studied. In the present paper the properties of the compact schemes constructed by a method of lines are studied. As the model problem for analyzing the scheme properties, the initial-boundary problem for a linear equation of a thermal conduction with discontinuous initial data is chosen. In a method of lines, the space derivative in the thermal conduction equation is approximated on a two-point stencil according to the formula of compact differentiation of the fourth order of accuracy. For solving an evolutionary system of ODEs various implicit one-step two-and three-stage schemes of the second and third order of accuracy are considered. Relation between properties of schemes stability functions and space monotonicity of numerical solution is analysed. Advantage of compact schemes in comparison with the traditional schemes using three-point approximating by a space derivative with the second order of accuracy in calculations on long time intervals is shown.