Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction‐diffusion equations

Abstract

On a vertical strip, a Dirichlet problem is considered for a system of two semilinear singularly perturbed parabolic reaction‐diffusion equations connected only by terms that do not involve derivatives. The highest‐order derivatives in the equations, having divergent form, are multiplied by the perturbation parameter e ; ϵ ∈ (0,1]. When ϵ → 0, the parabolic boundary layer appears in a neighbourhood of the strip boundary.

Using the integro‐interpolational method, conservative nonlinear and linearized finite difference schemes are constructed on piecewise‐uniform meshes in the x ₁ ‐axis (orthogonal to the boundary) whose solutions converge ϵ‐uniformly at the rate O (N1⁻² ln² N ₁ + N ₂ ⁻² + N ₀ ⁻¹). Here N ₁ + 1 and N ₀ + 1 denote the number of nodes on the x ₁‐axis and t‐axis, respectively, and N ₂ + 1 is the number of nodes in the x ₂‐axis on per unit length.

Key words:

• boundary value problem
• vertical strip
• system of semilinear equations
• parabolic reaction‐diffusion equations
• perturbation parameter ϵ
• parabolic boundary layer
• conservative difference schemes
• nonlinear and linearized difference schemes
• piecewise‐uniform mesh
• ϵ‐uniform convergence

Notes

This research was supported by the Russian Foundation for Basic Research under grants Nos. 07–01–00729, by the Boole Centre for Research in Informatics at the National University of Ireland in Cork, and by the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics initiative grant 06/MI/005.