Uniformly Convergent New Hybrid Numerical Method for Singularly Perturbed Parabolic Problems with Interior Layers

Yadav, Narendra Singh; Mukherjee, Kaushik

doi:10.1007/s40819-020-00804-7

Cited by 8 publications

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“…In the literature, several parameter-robust numerical methods (based on the fitted mesh methods) are developed for analyzing linear SPDEs with discontinuous data. To cite a few, for significant research contributions toward linear SPDEs mostly with discontinuous convection coefficient, one can recall the articles [2,[22][23][24][25][26][27] where solution of SPDE possesses strong interior layers and the articles [28][29][30][31] where solution of SPDE generates both boundary and strong interior layers. In this regard, we cite the recent research finding in Shiromani et al [32] for 2D elliptic singularly perturbed convection-diffusion problems with discontinuous convection and source terms, which give rise to strong interior layer phenomena.…”

Section: Introductionmentioning

confidence: 99%

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

Yadav,

Mukherjee

2024

Math Methods in App Sciences

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This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments.

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