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

Abstract

Abstract 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 𝒪 ⁢ ( 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 } , N 0 → ∞ {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in { ε ⁢ N } {\{\varepsilon N\}} and N 0 {N_{0}} ; as N , N 0 → ∞ {N,N_{0}\to\infty} , the convergence is conditional with respect to N, where N + 1 {N+1} and N 0 + 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 condition h ≤ 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 𝒪 ⁢ ( ε - 2 ⁢ N - 2 + N 0 - 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 (for ε = 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.