# Combined prefactored compact schemes for first- and second-order derivatives: conceptual derivation

## TL;DR

This paper presents a conceptual derivation of combined prefactored compact schemes that enhance the accuracy of first and second derivatives from sixth to tenth order without increasing stencil size, using Fourier analysis.

## Contribution

It introduces a novel derivation method for higher-order compact schemes that maintains computational efficiency while improving accuracy.

## Key findings

- Order of accuracy increased from sixth to tenth
- Same stencil size used for higher accuracy
- Closed set of equations for scheme weights derived

## Abstract

The derivation of combined prefactored compact schemes for first and second order derivatives is described here, relying on the Fourier analysis of the original prefactored compact schemes. By this approach, the order of accuracy of the original schemes can be increased from sixth to eight, or from eight to tenth (depending on the order of the original scheme), while the number of grid points in the stencil is kept the same. Here, we only frame the conceptual derivation of the schemes, leading to a closed set of equations for the weights.