Solving parabolic and hyperbolic equations by the generalized finite difference method.(English)Zbl 1139.35007
The numerical solution for parabolic and hyperbolic equations up to three space dimension problems is investigated in this paper.
The differential equations are of linear type with constant coefficients. As a numerical scheme generalized finite difference method is used. The results of this paper extent authors’ previous works in problem of numerical method based on generalized finite differences. For both, equations of parabolic and hyperbolic type they derived explicit finite difference formulae in which irregular clouds of grid points are used. Truncation errors for parabolic and hyperbolic equations is defined and proved. Then von Neumann stability criterion as a function of the coefficients of the star equation for irregular clouds of nodes is derived. These results present the extension of known stability results for the explicit method on regular grids for the more generalized case using irregular grids.
Different examples are included in the final section of the paper. One part is devoted to the problems with parabolic equation in 1, 2 and 3 space dimensions, in second part problems of hyperbolic equations are solved using proposed method. Computing the problems with exact solutions, the variation of global and maximum local error versus time steps are presented.
The differential equations are of linear type with constant coefficients. As a numerical scheme generalized finite difference method is used. The results of this paper extent authors’ previous works in problem of numerical method based on generalized finite differences. For both, equations of parabolic and hyperbolic type they derived explicit finite difference formulae in which irregular clouds of grid points are used. Truncation errors for parabolic and hyperbolic equations is defined and proved. Then von Neumann stability criterion as a function of the coefficients of the star equation for irregular clouds of nodes is derived. These results present the extension of known stability results for the explicit method on regular grids for the more generalized case using irregular grids.
Different examples are included in the final section of the paper. One part is devoted to the problems with parabolic equation in 1, 2 and 3 space dimensions, in second part problems of hyperbolic equations are solved using proposed method. Computing the problems with exact solutions, the variation of global and maximum local error versus time steps are presented.
Reviewer: Angela Handlovičová (Bratislava)
MSC:
| 35A35 | Theoretical approximation in context of PDEs |
| 35E20 | General theory of PDEs and systems of PDEs with constant coefficients |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 74S20 | Finite difference methods applied to problems in solid mechanics |
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References:
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