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Robust Computational Methods for Singularly Perturbed Advection-Subdiffusion Equations With Mild Temporal Singularities
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DOI：10.1002/num.70088.png)
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Abstract
En 中文
This work presents a comparative study of efficient computational methods for time-fractional singularly perturbed parabolic problems. The solutions to such problems are characterized by an initial layer near t=0, arising from the weak singularity of the fractional temporal derivative, and a boundary layer induced by the presence of a small perturbation parameter in the spatial operator. To accurately capture these multiscale features, three numerical schemes are developed and analyzed on layer-adapted meshes, highlighting the inadequacy of uniform meshes for this class of problems. In the L1-UP scheme, the fractional operator is approximated using the classical L1 method on a graded temporal mesh, while the spatial layer is resolved using an upwind difference discretization on a piecewise-uniform Shishkin mesh. The accuracy of this scheme is further enhanced via a post-processing extrapolation technique, leading to the L1-EXP scheme with improved convergence properties. To achieve higher spatial accuracy, a hybrid discretization combining midpoint upwind and central difference techniques is proposed in the L1-HYB scheme, yielding second-order accuracy in space. All proposed schemes are shown to be parameter-uniform and robust with respect to the perturbation parameter. Their performance is further assessed using a Bakhvalov-Shishkin mesh, which provides improved resolution within the boundary layer. Comprehensive numerical experiments are conducted to validate the theoretical convergence results and to illustrate the effectiveness of the derived methods in accurately resolving both initial and boundary layers.
Keywords:
advection-subdiffusion equation
boundary layer
convergence analysis
initial temporal singularity
singular perturbation
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