# The gradient discretisation method for linear advection problems

## TL;DR

This paper extends the Gradient Discretisation Method (GDM) to linear hyperbolic equations, unifying and analyzing various numerical schemes like finite elements and finite volumes for scalar advection problems.

## Contribution

It adapts GDM to hyperbolic equations, enabling unified design and convergence analysis of multiple numerical schemes for scalar advection.

## Key findings

- Convergence of the adapted GDM scheme is established.
- Numerical tests confirm the effectiveness of the method.
- The scheme accommodates various discretisation techniques.

## Abstract

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped P1 conforming and non conforming finite element and on the hybrid finite volume method.