Abstract In this work we investigate the hp-discontinuous Galerkin (DG) time-stepping method for the generalized Burgers–Huxley equation with memory, a nonlinear advection–diffusion–reaction problem featuring weakly singular kernels. We derive a priori error estimates for the semidiscrete scheme using hp-DG time-stepping, with explicit dependence on the local mesh size, polynomial degree and solution regularity, achieving optimal convergence in the energy norm. For the fully discrete scheme we initially implement the hp-conforming finite element method, followed by the hp-DG method. We establish the well-posedness and stability of the fully discrete scheme and provide corresponding a priori estimates. The effectiveness of the proposed method is demonstrated through numerical validation on a series of test problems.
Mahajan et al. (Tue,) studied this question.