On the consistent two-side estimates for the solutions of quasilinear convection-diffusion equations and their approximations on non-uniform grids

A new second-order in space linearized difference scheme on non-uniform grid that approximates the Dirichlet problem for multidimensional quasilinear convection-diffusion equation with unbounded nonlinearity is constructed. Proposed algorithm is a novel nonlinear generalization of difference schemes for linear problems developed earlier by the authors. Nontrivial two-side pointwise estimates of the solution of the scheme fully consistent with the corresponding estimates for the differential problem are established. Such estimates permit to prove the nonnegativity of the exact solution, important in physical problems, and also to find sufficient conditions on the input data when the nonlinear problem is parabolic. As a result a priori estimates of the approximate solution in the grid norm C that depend on the initial and boundary conditions and on the right-hand side only are proved.