# Exact finite difference schemes for three-dimensional linear systems with constant coefficients

## TL;DR

This paper develops implicit and explicit exact difference schemes for three-dimensional linear systems with constant coefficients, demonstrating their efficiency and accuracy through numerical simulations, and extends the approach to more general systems.

## Contribution

It introduces new exact difference schemes for 3D linear systems and shows their effectiveness compared to high-order methods, with potential extensions to nonlinear systems.

## Key findings

- Numerical simulations confirm efficiency for stiff problems.
- Exact difference schemes outperform high-order methods.
- Potential extension to nonlinear systems with stability preservation.

## Abstract

In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for problems with periodic solutions on very large time interval demonstrate the efficiency and exactness of the EDS compared with high-order numerical methods. This result can be extended for constructing EDS for general systems of $n$ linear differential equations with constant coefficients and nonstandard finite difference (NSFD) schemes preserving stability properties for quasi-linear system of equations $\textbf{x}' = A\textbf{x }+ f(\textbf{x})$.