# Robust Numerical Solution for Solving Elastohydrodynamic Lubrication (EHL) Problems using Total Variation Diminishing (TVD) Approach

## TL;DR

This paper introduces a stable TVD scheme with hybrid line splittings for solving complex elastohydrodynamic lubrication problems efficiently, reducing computational complexity and handling high-order discretizations without perturbing robustness.

## Contribution

It presents a novel TVD-based numerical method with hybrid line splittings tailored for EHL problems, improving efficiency and robustness over existing schemes.

## Key findings

- Reduces computational complexity to O(n log n)
- Handles high-order discretizations directly
- Maintains robustness across a wide load parameter range

## Abstract

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).