# A stable semi-implicit algorithm

## TL;DR

This paper introduces a semi-implicit splitting algorithm that ensures stability in numerical integration when the evolution operator's singular spectrum mixes values greater and less than one, demonstrated on a tokamak model.

## Contribution

A novel semi-implicit splitting method that guarantees stability for mixed-spectrum evolution operators in complex systems.

## Key findings

- Successfully applied to a tokamak scrape-off layer model
- Ensures stability where explicit and implicit methods fail
- Provides a practical solution for mixed-spectrum stability issues

## Abstract

When the singular values of the evolution operator are all smaller or all greater than one, stable integration algorithms are obtained either by explicit or implicit methods. When the singular spectrum mixes greater and smaller than one values, neither explicit nor implicit methods insure stabilty. The problem is solved by using a splitting of the evolution operator and a semi-implicit scheme. The method is illustrated in the study of a two-field model of the tokamak scrape-off layer.

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Source: https://tomesphere.com/paper/1905.04520