A decomposition method for singularly perturbed parabolic convection-diffusion equations with discontinuous initial conditions (Q2714027)

The author studies the Cauchy problem for a singularly perturbed parabolic equation of the form NEWLINE\[NEWLINELu(x,t) = f(x,t),\quad (x,t)\in G,\qquad u(x,t) = \varphi(x),\quad (x,t)\in S,NEWLINE\]NEWLINE where \(G = \{(x,t): x\in\mathbb R, \;t\in (0,T]\}\), \(S=\overline G\setminus G\), \(Lu(x,t) = L^{2}u(x,t) + L^{1}u(x,t)\) and NEWLINE\[NEWLINE \begin{aligned} L^{2}u(x,t) &= \varepsilon^2\left\{a(x,t)\frac{\partial^2}{\partial x^2} + b_0(x,t)\frac{\partial}{\partial x} - c_0(x,t)\right\}u(x,t), \\ L^{1}u(x,t) &= \left\{-c(x,t) - p(x,t)\frac{\partial}{\partial t}\right\} u(x,t). \end{aligned} NEWLINE\]NEWLINE Sufficient smoothness and boundedness of the coefficients in \(\overline G\) are assumed. The initial function \(\varphi(x)\) has first-order discontinuity on the set \(S^* = \{(x,t):x= 0\), \(t = 0\}\) and is sufficiently smooth and bounded on \(S\setminus S^*\).NEWLINENEWLINENEWLINEThe aim of the article is to construct special difference schemes that \(\varepsilon\)-uniform converge on the whole grid domain. The main idea of this construction is connected with the use of the domain and solution decomposition technique. The singular solution which is generated by discontinuity of the initial data is extracted in an explicit form in a neighborhood of discontinuity and, next, the author employes the meshes condensing in a special way in a neighborhood of the transient layer.