FINITE DIFFERENCE SCHEMES IN A MULTI-POINT STENCIL AND FINITE DIFFERENCE SCHEMES WITH THE EXACT SPECTRUM FOR PERIODIC BOUNDARY CONDITIONS
Abstract
In this paper the finite difference scheme (FDS) for local approximation of periodic function's derivatives in a 2n+1 point stencil is studied, obtaining higher order accuracy approximation. This method in the uniform grid with N mesh points is used to approximate the differential operator of the second and the first order derivatives in the space, using the multi-point stencil. The matrix representation A of this FDS is investigated also for other order derivatives. It is shown that the eigenvalues of FDS matrix representation A can be obtained as a sum whose terms do not depend on n. This allows easily solving the FDS by the spectral decomposition of A.
The described methods are applicable for various mathematical physics problems involving periodic boundary conditions (PBC). The solutions of some problems of parabolic type partial differential equations (PDE) with PBCs are obtained, using the method of lines (MOL) to approach the PDEs in the time and the discretization in space applying the FDS of a different order of the approximation and finite difference scheme with exact spectrum (FDSES).
These methods are compared with the global approximations or pseudospectral methods, which are based on using differentiation matrices (DMs) for derivatives on a uniform grid with trigonometric interpolation.\pagebreak
In this paper we show that the approximation FDS in the limit case when n tends to infinity is equivalent to FDSES; moreover, FDSES is equivalent to a spectral differentiation matrix based on trigonometric (Fourier) interpolant.
The described methods are applicable for various mathematical physics problems involving periodic boundary conditions (PBC). The solutions of some problems of parabolic type partial differential equations (PDE) with PBCs are obtained, using the method of lines (MOL) to approach the PDEs in the time and the discretization in space applying the FDS of a different order of the approximation and finite difference scheme with exact spectrum (FDSES).
These methods are compared with the global approximations or pseudospectral methods, which are based on using differentiation matrices (DMs) for derivatives on a uniform grid with trigonometric interpolation.\pagebreak
In this paper we show that the approximation FDS in the limit case when n tends to infinity is equivalent to FDSES; moreover, FDSES is equivalent to a spectral differentiation matrix based on trigonometric (Fourier) interpolant.
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