Impact of mixed boundary conditions and nonsmooth data on layer-originated nonpremixed combustion problems: Higher-order convergence analysis

This work explores the theoretical and computational impacts of mixed-type flux conditions and nonsmooth data on boundary/interior layer-originated singularly perturbed semilinear reaction–diffusion problems. Such problems are prevalent in nonpremixed combustion models and catalytic reaction models. The inclusion of arbitrarily small diffusion terms results in boundary layers influenced by specific flux conditions normalized by perturbation parameters. We have demonstrated theoretically that the sharpness of the boundary layer is significantly reduced when this normalization is independent of the diffusion parameter. In addition, the presence of a nonsmooth source function gives rise to an interior layer in the current problem. We show that using upwind discretizations for mixed boundary fluxes achieves nearly second-order accuracy if the first derivatives are not normalized concerning perturbation parameters. This outcome arises from the bounds of a prior derivative of the continuous solution. Furthermore, it is theoretically shown that nearly second-order uniform accuracy can be attained for reaction-dominated semilinear problems using a hybrid scheme at the discontinuity point. To ensure the uniform stability of the discrete solution, a transformation is necessary for the corresponding discrete problem. Theoretical results are supported by various experiments on nonlinear problems, illustrating the pointwise rates and highlighting both linear and higher-order accuracy at specific points.