Open AccessThe paper concerns a task of selecting the spatial (mesh) and temporal (time step) discretization in problems to be solved by finite volume method.
The method is proposed in which the choice of the case settings is directly based on the analysis of a discretization matrix. The papers [2, 3, 4, 5] deal with the methods based on the analysis of the gradient field or value field of physical quantities. Papers [6], [7], [8] describe methods for a posteriori estimation of the so-called anisotropic interpolation errors and methods for creating an optimal triangular mesh [9], [10].
A common disadvantage of these methods is that it is necessary to solve the problem to obtain the output values. In this paper we propose a procedure based on the relationship of the discretization matrix multicollinearity with a mesh, a time step, and a chosen set of finite difference schemes.
A problem with a simple geometry in which the flow is specified by the hyperbolic equation, is chosen as an example. A number of experiments have been conducted to compare variations on the meshes for the specified difference schemes.
A type of the elements of considered meshes is hexahedrons. Meshes both with constant and with variable spatial step were used.
Calculations based on the normalized discretization matrix were performed for each variation of mesh to determine index of diagonal dominance, condition number, determinant, and minimum eigenvalue. To obtain multicollinearity, a diagonal dominance index was used. The paper shows the relationship of this parameter with the condition of solvable linear algebraic equation.
The computational complexity of the abovementioned index of diagonal dominance is significantly lower as compared to indices of condition such as a condition number and a minimum eigenvalue.
This allows us to estimate the condition of the linear algebraic equation with the least computational cost. The proposed procedure allows comparison and selection of the most appropriate spatial (mesh) and temporal (time step) discretization without a complete trial solution of the linear algebraic equation, thereby significantly reducing the computational complexity of solving problems.