Abstract: Some asymptotical behaviours in solution of mixed initial-boundary value problems of quasi-linear equations as t→∞ are discussed in this paper. It is shown that the solution of the system of non-steady equations with steady boundary conditions tends to the solution of the steady system with the same boundary conditions as t→∞, provided that the solution of the steady problem exists and the solution of the non-steady problem is unique. The exact computational method for some quantities at the boundary points is also considered when a finite-difference method is applied to solving this kind of problems.In order to get rid of the local non-physical solution in the vicinity of a sharp edge on the body, and get correct solution of boundary value problems about concave bodies, a simple boundary scheme of dissipative type is given. some numerical examples of concave bodies with sharp corners are presented.