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On the finite difference approximation to the convection-diffusion equation. (English) Zbl 1100.65070
Summary: The system of ordinary differential equations arisen from discretizing the convection-diffusion equation with respect to the space variable to compute its approximate solution along a time level is considered. This system involves in computing $e^{k\bold A}y$ for some vector $y$, where $k$ is the time step-size and $\bold A$ is a large tridiagonal Toeplitz matrix. The common ways to compute an approximate solution of the convection-diffusion equation are based on replacing $e^{k\bold A}y$ by an its approximation. We give an explicit expression for the exact value of $e^{k\bold A}y$ and then the numerical results of our method with that of some well-known methods are compared.

##### MSC:
 65M06 Finite difference methods (IVP of PDE) 65M20 Method of lines (IVP of PDE) 35K15 Second order parabolic equations, initial value problems 65F10 Iterative methods for linear systems
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##### References:
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