# A linear implicit finite difference discretization of the Schrodinger-Hirota Equation

## TL;DR

This paper introduces a linear implicit finite difference method for solving the Schrödinger-Hirota equation, achieving second-order convergence and verified through numerical experiments, improving computational efficiency for periodic initial value problems.

## Contribution

The paper presents a novel linear implicit finite difference scheme with proven second-order convergence for the Schrödinger-Hirota equation, validated by numerical experiments.

## Key findings

- Achieved optimal second-order convergence in the discrete H^1-norm.
- Validated the efficiency of the method through numerical experiments.
- Provided conditions for the stability and accuracy of the scheme.

## Abstract

A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved, assuming that $\tau$, $h$ and $\tau^4/h$ are sufficiently small, where $\tau$ is the time-step and $h$ is the space mesh-size. The efficiency of the proposed method is verified by results from numerical experiments.