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ABSTRACT
In this article, a new unconditionally stable scheme, based on the Associated Hermite orthogonal functions combined with first-order upwind scheme (AH-FUS), is proposed for solving the convection–diffusion equation. To eliminate the time variable from the computations, the time derivatives are expanded by Hermite functions, and a Galerkin's temporal testing procedure is introduced to the expanded equation. A set of implicit difference equations are derived in AH domain under no convergent conditions, and the numerical results can be obtained by solving the expanded coefficients recursively. Two numerical examples were considered to verify the accuracy and the efficiency of the proposed scheme.
Nomenclature
| a | = | diffusion coefficient |
| A | = | banded matrix [Eq. (16)] |
| c | = | convection coefficient |
| H | = | Hermite polynomial |
| I | = | unit matrix |
| S | = | heat flux |
| t | = | time |
| = | transformed time variable | |
| T | = | time interval |
| Tf | = | time translating parameter |
| u | = | solution of equation |
| α | = | tridiagonal matrix defined in Eq. (15) |
| β | = | unit matrix defined in Eq. (15) |
| λ | = | matrix coefficient [Eq. (15)] |
| σ | = | time-scaling parameter |
| ϕ | = | Associated Hermite basis function |
| Ω | = | computational domain |
| Subscripts and Superscripts | = | |
| i, j | = | value at grid spatial points |
| k, m, n, p | = | order of expansion coefficient |
| Nx | = | number of grid points in x direction |
| Ny | = | number of grid points in y direction |
Nomenclature
| a | = | diffusion coefficient |
| A | = | banded matrix [Eq. (16)] |
| c | = | convection coefficient |
| H | = | Hermite polynomial |
| I | = | unit matrix |
| S | = | heat flux |
| t | = | time |
| = | transformed time variable | |
| T | = | time interval |
| Tf | = | time translating parameter |
| u | = | solution of equation |
| α | = | tridiagonal matrix defined in Eq. (15) |
| β | = | unit matrix defined in Eq. (15) |
| λ | = | matrix coefficient [Eq. (15)] |
| σ | = | time-scaling parameter |
| ϕ | = | Associated Hermite basis function |
| Ω | = | computational domain |
| Subscripts and Superscripts | = | |
| i, j | = | value at grid spatial points |
| k, m, n, p | = | order of expansion coefficient |
| Nx | = | number of grid points in x direction |
| Ny | = | number of grid points in y direction |
Acknowledgment
This work was supported by PLA University of Science and Technology.