Open AccessCONSERVATIVE NUMERICAL METHOD FOR A SYSTEM OF SEMILINEAR SINGULARLY PERTURBED PARABOLIC REACTION‐DIFFUSION EQUATIONS
Author(s) -
Lidia P. Shishkina,
Г. И. Шишкин
Publication year - 2009
Publication title -
mathematical modelling and analysis/mathematical modeling and analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.491
H-Index - 25
eISSN - 1648-3510
pISSN - 1392-6292
DOI - 10.3846/1392-6292.2009.14.211-228
Subject(s) - mathematics , reaction–diffusion system , mathematical analysis , singular perturbation , dirichlet boundary condition , perturbation (astronomy) , nonlinear system , parabolic partial differential equation , boundary (topology) , piecewise , partial differential equation , physics , quantum mechanics
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.