Abstract
We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.
Highlights
Approximation of the di erential problem is based on the conservation law approach of the nite volumes method 4]
Important to work out special methods of solution, the so-called uniform converging computational methods.This gives a basis for the development of special monotone nite vector-di erence schemes with perturbation coe cient of function-matrix for solving the system of di erential equations
Special nite-di erence approximations are constructed for a steady-state boundary-value problem, systems of parabolic type partial di erential equations, a system of two MHD equations, 2-D ows and MHD- ows equations in curvilinear orthogonal coordinates
Summary
Approximation of the di erential problem is based on the conservation law approach of the nite volumes method 4]. We will consider the exact nite-di erence scheme for solving the system of di erential equations of second order with piece-wise constant coe cients. The presence of of the the matrix-functions matrix a and vge(cst)orrs(sf) in the ensure case of piece-wise constant elements the exact discrete approximation of corresponding 1 - D boundary value problem.
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