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<p>This article develops and analyzes a fast robustly convergent numerical method to solve singularly perturbed parabolic convection-diffusion systems, which have a non-smooth source term. The diffusion parameters, which multiply the highest order derivative at each equation of the system, are allowed to be distinct and possibly of varying magnitudes. Then, in general, for sufficiently small values of the diffusion parameters, we observe overlapping boundary layer phenomenon in the exact solution of the continuous problem. Moreover, these boundary layers get accompanied with overlapping interior layers when there is a non-smooth source term in the problem. We establish a priori bounds on the solution and its derivatives, and also a decomposition of the solution into a smooth part, a boundary layer part and an interior layer part. To approximate the solution, we construct a fast numerical method, which uses an upwind finite difference scheme, defined on an appropriate Shishkin type mesh in space and the fractional implicit Euler method combined with the components-wise splitting approach, to discretize in time, on an evenly spaced time grid. This in turn implies that the approximation of the components of the numerical solution become decoupled, permitting us to handle only tridiagonal linear systems of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N_{x}+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>x</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> at each discrete time step. Further, due to overlapping boundary and interior layers the error analysis is challenging and requires the construction of novel barrier functions. It is proven that the proposed numerical method is uniformly convergent with first order accuracy in time and essentially first order in space. Numerical outcomes are presented in the form of tables alongside with a comparison of computational times for various test examples which corroborate in practice the theoretical results. The comparison distinctly illustrates the efficiency of our proposed algorithm.</p>

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A fast robust numerical method for singularly perturbed parabolic convection-diffusion systems with a non-smooth source term

  • 期刊论文

  • 发表日期：

    2026年3月3日

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  • Aishwarya Jaiswal,
  • Sunil Kumar,
  • Carmelo Clavero

摘要

This article develops and analyzes a fast robustly convergent numerical method to solve singularly perturbed parabolic convection-diffusion systems, which have a non-smooth source term. The diffusion parameters, which multiply the highest order derivative at each equation of the system, are allowed to be distinct and possibly of varying magnitudes. Then, in general, for sufficiently small values of the diffusion parameters, we observe overlapping boundary layer phenomenon in the exact solution of the continuous problem. Moreover, these boundary layers get accompanied with overlapping interior layers when there is a non-smooth source term in the problem. We establish a priori bounds on the solution and its derivatives, and also a decomposition of the solution into a smooth part, a boundary layer part and an interior layer part. To approximate the solution, we construct a fast numerical method, which uses an upwind finite difference scheme, defined on an appropriate Shishkin type mesh in space and the fractional implicit Euler method combined with the components-wise splitting approach, to discretize in time, on an evenly spaced time grid. This in turn implies that the approximation of the components of the numerical solution become decoupled, permitting us to handle only tridiagonal linear systems of order \(N_{x}+1\) N x + 1 at each discrete time step. Further, due to overlapping boundary and interior layers the error analysis is challenging and requires the construction of novel barrier functions. It is proven that the proposed numerical method is uniformly convergent with first order accuracy in time and essentially first order in space. Numerical outcomes are presented in the form of tables alongside with a comparison of computational times for various test examples which corroborate in practice the theoretical results. The comparison distinctly illustrates the efficiency of our proposed algorithm.