In this paper, a higher order nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov mesh is developed for a weakly coupled system of singularly perturbed reaction-diffusion equations. At first, by selecting special penalty parameters at different mesh points, the supercloseness of the $k+\frac{1}{2}$ order of our proposed method is derived, where $k$ is the degree of polynomials space. Then, an optimal order of uniform convergence analysis in a balanced norm is performed. Finally, some numerical experiments are given to support our theoretical findings.

An adaptive moving grid method is developed to solve the time-fractional integrodifferential governing equations of viscoelastic nanofluid. The momentum equation is derived based on a dual-parameter fractional Maxwell constitutive relation, and the energy equation employs a generalized Cattaneo heat conduction relation. To improve solution accuracy, a monitor function based on the equidistribution principle is constructed, and an adaptive mesh redistribution strategy is developed in the spatial domain. The temporal fractional-order operators are approximated by the L1 algorithm and the weighted-shifted Grünwald difference method. Numerical experiments demonstrate that the adaptive grid achieves 77.6–88.4% higher accuracy compared to uniform grids at the same grid scale, along with enhanced stability in convergence. Parametric analysis indicates that increasing the fractional-order derivative in the energy equation results in a thickening of both the velocity and thermal boundary layers. Furthermore, the dual-fractional Maxwell model exhibits a thicker velocity boundary layer than its classical single-parameter counterpart. The proposed method offers an efficient and robust approach for simulating complex viscoelastic nanofluid systems with memory effects and multi-field coupling.

In this paper, a second finite difference method on a graded grid is proposed for a Volterra integro-differential equation with a weakly singular kernel. The proposed scheme is obtained by using the two-step backward differentiation formula (BDF2) to discretize the first derivative term and the first-order interpolation scheme to approximate the integral term. The analysis of stability is proved and used to prove the convergence of our presented numerical method in the discrete maximum norm. Finally, Numerical experiments are given to verify the theoretical results.