Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature
The paper studies the numerical solution of the following singularly perturbed system of reaction-diffusion two-point boundary-value problems: \[ \begin{array}{ll} \left( \begin{array}{cc} -\varepsilon\dfrac{d^2}{dx^2} & 0 \\[6pt] 0 & -\mu\dfrac{d^2}{dx^2} \end{array}\right) \left( \begin{array}{c} u_1(x) \\[6pt] u_2(x) \end{array}\right) + \left( \begin{array}{cc} b_{11}(x) & b_{12}(x) \\[6pt] b_{21}(x) & b_{22}(x) \end{array}\right) \left( \begin{array}{c} u_1(x) \\[6pt] u_2(x) \end{array}\right) = \left( \begin{array}{c} f_1(x) \\[6pt] f_2(x) \end{array}\right) \end{array}, \] subject to one of the following conditions: \begin{itemize} \item[(i)] Dirichlet boundary conditions: \[ u_1(0) = u_1(1) = u_2(0) = u_2(1) = 0, \] \item[(ii)] Mixed boundary conditions: \[ \left\{ \begin{array}{ll} \alpha_1 u_1(0) - \beta_1 \sqrt{\varepsilon}u_1'(0) = A_1, \quad \gamma_1 u_1(1) + \delta_1 \sqrt{\varepsilon}u_1'(1) = B_1, \\[4pt] \alpha_2 u_2(0) - \beta_2 \sqrt{\mu}u_2'(0) = A_2, \quad \gamma_2 u_1(1) + \delta_2 \sqrt{\mu}u_2'(1) = B_2. \end{array}\right. \] \end{itemize} Under suitable assumptions on the coefficients in the equations, and in the boundary conditions, the system admits a unique solution exhibiting boundary layers at \(x=0\) and \(x=1\). Here, the authors apply the classical finite difference schemes to discretize the differential equations, and for the mixed boundary conditions, they use both the classical forward and backward difference schemes. Moreover cubic spline based difference schemes are used on layer-adapted nonuniform grids obtained through equidistribution principle with a suitable monitor function. Parameter-uniform error estimates of the first and second orders are obtained and numerical results are carried out to show the applicability of the method. The numerical results reveal the theoretical order of convergence. For comparison of the numerical results, they apply the present schemes on both the uniform and Shishkin meshes. There is no need to apply the scheme on the uniform mesh, because it is well-known that uniform mesh will not give uniform convergence result.
scientific article; zbMATH DE number 7137379
scientific article