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

• Main Authors: Lidia Shishkina, Grigorii Shishkin
• Format: Article
• Language: English
• Published: Vilnius Gediminas Technical University 2009-06-01
• Series: Mathematical Modelling and Analysis
• Subjects: boundary value problem; vertical strip; system of semilinear equations; parabolic reaction‐diffusion equations; perturbation parameter ϵ; parabolic boundary layer
• Online Access: https://journals.vgtu.lt/index.php/MMA/article/view/6541

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. First published online: 14 Oct 2010