A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations
The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives along y-direction are remo...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/457938 |
| Summary: | The idea of direction changing and order reducing is proposed to generate an exponential
difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion
equation with source term. During the derivation process, the higher order derivatives along y-direction are removed to the derivatives along x-direction iteratively
using information given by the original differential equation (similarly from x-direction to
y-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite
terms which constitute convergent series are kept on deriving the exponential coefficients
of the scheme. From the construction process one may gain more insight into the relations
among the stencil coefficients. The scheme is of positive type so it is unconditionally
stable and the convergence rate is proved to be of second-order. Fourth-order accuracy
can be obtained by applying Richardson extrapolation algorithm. Numerical results
show that the scheme is accurate, stable, and especially suitable for convection-dominated
problems with different kinds of boundary layers including elliptic and parabolic ones.
The idea of the method can be applied to a wide variety of differential equations. |
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| ISSN: | 1110-757X 1687-0042 |