Analysis of an almost fourth-order parameter-uniformly convergent numerical method for singularly perturbed semilinear reaction-diffusion system with non-smooth source term

Highlights：

• A one dimensional singularly perturbed semilinear reaction-diffusion system with two or more equations that has a discontinuity at a point in the source term is examined numerically.

• The presence of a discontinuity in the source term generally causes the reduction of the desired order of convergence in any numerical technique applied to the problem.

• The existence of a solution to the continuous problem has been proved using the concept of upper and lower solutions. The considered semilinear problem is lin- earized, and the Maximum principle has been proved for the linearized problem.

• A combination of generalized Numerov scheme and classical central difference scheme is used at the nodal point, which is not a point of discontinuity. A special finite difference scheme is constructed at the point of discontinuity. The scheme uses an appropriate generalized Shishkin mesh fitted to the interior and boundary layers.

• The decomposition of the exact and numerical solutions are made to prove al- most fourth-order parameters-uniform convergence for the approximations gen- erated by the finite difference scheme in the discrete maximum norm.

• To our knowledge, the present result is the state of art result for the problem under consideration.

摘要

•A one dimensional singularly perturbed semilinear reaction-diffusion system with two or more equations that has a discontinuity at a point in the source term is examined numerically.•The presence of a discontinuity in the source term generally causes the reduction of the desired order of convergence in any numerical technique applied to the problem.•The existence of a solution to the continuous problem has been proved using the concept of upper and lower solutions. The considered semilinear problem is lin- earized, and the Maximum principle has been proved for the linearized problem.•A combination of generalized Numerov scheme and classical central difference scheme is used at the nodal point, which is not a point of discontinuity. A special finite difference scheme is constructed at the point of discontinuity. The scheme uses an appropriate generalized Shishkin mesh fitted to the interior and boundary layers.•The decomposition of the exact and numerical solutions are made to prove al- most fourth-order parameters-uniform convergence for the approximations gen- erated by the finite difference scheme in the discrete maximum norm.•To our knowledge, the present result is the state of art result for the problem under consideration.