# Stability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations

## TL;DR

This paper analyzes the stability of explicit Runge-Kutta methods in high order finite element approximations of linear parabolic equations, providing bounds on the system matrix eigenvalues that influence time step size.

## Contribution

It introduces new bounds on the eigenvalues of the system matrix, linking stability to mesh geometry, basis functions, and diffusion matrix properties.

## Key findings

- Bounds on eigenvalues depend on matrix entry ratios and mesh geometry.
- The tightness of bounds varies with problem parameters and mesh configuration.
- Results offer insights into stability constraints for explicit schemes on nonuniform meshes.

## Abstract

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.