| Abstract: |
| This research presents an advanced numerical technique designed for solving two-dimensional singularly perturbed semilinear parabolic convection-diffusion equations, characterized by time-dependent non-homogeneous boundary conditions. The proposed method is a combination of an implicit-explicit fitted mesh method (FMM) and a Richardson extrapolation approach. The temporal discretization is handled through an Alternating Direction Implicit-Explicit (ADI) Euler scheme, which ensures accurate handling of time-dependent boundary values. For spatial discretization, we employ a hybrid finite difference scheme on a non-uniform rectangular mesh, while the time domain is discretized using a uniform grid.
To begin, the paper explores the stability and asymptotic behavior of the analytical solution for the nonlinear problem. Following this, the stability properties of the implicit-explicit method are examined, and the convergence of the numerical solution is established, showing that the method achieves uniform convergence with respect to the perturbation parameter $\varepsilon$. The Richardson extrapolation technique is applied specifically to the time variable to enhance the accuracy and order of convergence in the temporal dimension.
The proposed method is validated through a series of numerical experiments that confirm the theoretical predictions regarding stability and convergence rates. These numerical results demonstrate the effectiveness of the method in handling complex boundary conditions and solving the nonlinear parabolic PDEs with high accuracy. |
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