English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest  x -derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval(0,1]{(0,1]}. For small ε, the problem involves a boundary layer of width⁢(ε){\mathcal{O}(\varepsilon)}, where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as{ε⁢N},N0→∞{\{\varepsilon N\},N_{0}\to\infty}in the maximum norm with first-order accuracy in{ε⁢N}{\{\varepsilon N\}}andN0{N_{0}}; asN,N0→∞{N,N_{0}\to\infty}, the convergence is  conditional with respect to   N , whereN+1{N+1}andN0+1{N_{0}+1}are the numbers of mesh points in  x  and  t , respectively. We develop an  improved  difference scheme on uniform meshes using the grid approximation of the first  x -derivative in the convective term by the  central  difference operator  under the conditionh≤m⁢ε{h\leq m\varepsilon}, which ensures the monotonicity of the scheme; here  m  is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a  rate  of⁢(ε-2⁢N-2+N0-1){\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})}. We compare the convergence rate of the developed scheme with the known Samarskii scheme for a  regular  problem. It is found that the improved scheme (forε=1{\varepsilon=1}), as well as the Samarskii scheme, converges  in the maximum norm  with  second-order accuracy in   x  and  first-order accuracy in   t .

Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation