# Operator-difference schemes on non-uniform grids for second-order evolutionary equations

## TL;DR

This paper develops unconditionally stable, second-order accurate operator-difference schemes on non-uniform grids for second-order evolutionary equations, ensuring stability and accuracy in numerical solutions with variable time steps.

## Contribution

It introduces new unconditionally stable, second-order schemes on non-uniform grids for second-order evolution equations, using a transformation to first-order systems and three-level schemes.

## Key findings

- Schemes are unconditionally stable and second-order accurate.
- Numerical experiments confirm stability and accuracy on non-uniform grids.
- The approach effectively handles variable time steps in evolutionary equations.

## Abstract

The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring the approximate solution's stability. In this paper, we construct unconditionally stable first- and second-order accuracy schemes on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.