| Citation: |
Narendra Singh Yadav, Kaushik Mukherjee. HIGHER-ORDER UNIFORM CONVERGENCE AND ORDER REDUCTION ANALYSIS OF A NOVEL FRACTIONAL-STEP FMM FOR SINGULARLY PERTURBED 2D PARABOLIC PDES WITH TIME-DEPENDENT BOUNDARY DATA[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1222-1268. doi: 10.11948/20230023
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HIGHER-ORDER UNIFORM CONVERGENCE AND ORDER REDUCTION ANALYSIS OF A NOVEL FRACTIONAL-STEP FMM FOR SINGULARLY PERTURBED 2D PARABOLIC PDES WITH TIME-DEPENDENT BOUNDARY DATA
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Abstract
The aim of this paper is to develop and analyze a cost-effective high-order efficient numerical method for a class of two-dimensional singularly perturbed parabolic convection-diffusion problems with non-homogeneous time-dependent boundary data. To achieve the goal, we develop an efficient fractional-step fitted mesh method that combines the fractional implicit-Euler method with an alternative evaluation of the boundary data for discretization in time, and consists of a new hybrid finite difference method for discretization in space. In the case of fully discrete numerical approximation of evolutionary PDEs, in particular, with time-dependent boundary conditions; it is noticed that the classical evaluation of the boundary data usually causes the order reduction in time; and it becomes severe when the fractional-step method is used. In this regard, the novelty of the current algorithm, other than its higher-order accuracy, is that the method can eliminate the order-reduction in time before and after the extrapolation with an appropriate evaluation of the boundary data; and at the same time, it can reduce the computational cost for solving the multi-dimensional problem by using the fractional-step method that converts the multi-dimensional system into two-independent one-dimensional subsystems at each time level. To accomplish this, we discretize the spatial domain using a layer-adapted non-uniform rectangular mesh and the time domain by an equidistant mesh. Stability and $ \varepsilon $-uniform convergence result of the fully discrete scheme are established in the supremum-norm. Moreover, the Richardson extrapolation technique is implemented solely in the time direction to enhance the order of convergence in time. Finally, numerical results are presented with the default and alternative choice for the boundary data to validate the theoretical findings.
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References
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Narendra Singh Yadav, Kaushik Mukherjee. HIGHER-ORDER UNIFORM CONVERGENCE AND ORDER REDUCTION ANALYSIS OF A NOVEL FRACTIONAL-STEP FMM FOR SINGULARLY PERTURBED 2D PARABOLIC PDES WITH TIME-DEPENDENT BOUNDARY DATA[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1222-1268. doi: 10.11948/20230023
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Figure 1.
Shishkin mesh in the spatial direction
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Figure 2.
Graphs of numerical solution for Example 6.1
$ N=64,\; M=32 $ $ t=1 $ -
Figure 3.
Graphs of numerical solution for Example 6.2
$ N=64,\; M=32 $ $ t=1 $ -
Figure 4.
Graphs of error
$ |U^{N,\Delta t}-U^{2N,\Delta t/2}| $ $ \varepsilon=2^{-6} $ $ \varepsilon=2^{-14} $ $ t=1 $ -
Figure 5.
Graphs of error
$ |U^{N,\Delta t}-u| $ $ \varepsilon=2^{-6} $ $ \varepsilon=2^{-14} $ $ t=1 $ -
Figure 6.
Loglog plot for comparison of the
$ \varepsilon $ $ \widehat{e}^{N,\Delta t} $ -
Figure 7.
Loglog plot for comparison of the
$ \varepsilon $ $ e^{N,\Delta t} $