# Streamline integration as a method for structured grid generation in X-point geometry

## TL;DR

This paper presents a method for generating structured grids aligned with flux-functions in X-point geometries, using streamline integration and grid refinement to improve numerical solutions of elliptic equations.

## Contribution

It introduces a novel grid construction algorithm based on flux-function properties and demonstrates how to restore convergence rates through local grid refinement.

## Key findings

- Orthogonal grids exist if Laplacian of flux-function vanishes at X-point.
- Grid refinement near X-point improves convergence of elliptic solutions.
- Proposed method enables structured grid generation in complex X-point geometries.

## Abstract

We investigate structured grids aligned to the contours of a two-dimensional flux-function with an X-point (saddle point). Our theoretical analysis finds that orthogonal grids exist if and only if the Laplacian of the flux-function vanishes at the X-point. In general, this condition is sufficient for the existence of a structured aligned grid with an X-point. With the help of streamline integration we then propose a numerical grid construction algorithm. In a suitably chosen monitor metric the Laplacian of the flux-function vanishes at the X-point such that a grid construction is possible. We study the convergence of the solution to elliptic equations on the proposed grid. The diverging volume element and cell sizes at the X-point reduce the convergence rate. As a consequence, the proposed grid should be used with grid refinement around the X-point in practical applications. We show that grid refinement in the cells neighboring the X-point restores the expected convergence rate.