  In this study, we propose a class of total variation diminishing (TVD)
schemes for solving pseudo-monotone variational inequality arises in
elasto-hydrodynamic lubrication point contact problem. A limiter based stable
hybrid line splittings are introduced on hierarchical multi-level grid. These
hybrid splittings are designed by use of diffusive coefficient and mesh
dependent switching parameter in the computing domain of interest. The spectrum
of illustrated splittings is derived with the help of well known local Fourier
analysis (LFA). Numerical tests validate the performance of scheme and its
competitiveness to the previous existing schemes. Advantages of proposed
splittings are observed in the sense that it reduces computational complexity
(up to ($O(n\log n)$) and solve high order discretization directly (no
defect-correction tool require) without perturbing the robustness of the
solution procedure (i.e. it works well for large range of load parameters).
