Error estimates for splitting methods based on AMF-Runge-Kutta formulas for the time integration of advection diffusion reaction PDEs

TL;DR

This paper analyzes the convergence and error estimates of AMF-Runge-Kutta methods, especially Radau IIA based schemes, for efficiently solving semi-discretized advection-diffusion-reaction PDEs with proven error bounds and numerical validation.

Contribution

It introduces error estimates for AMF-RK methods applied to PDEs, demonstrating their efficiency and accuracy with theoretical bounds and numerical experiments.

Findings

AMF-RK methods achieve competitive accuracy for multidimensional PDEs.

Uniform error bounds are established for combined space-time discretizations.

Numerical results confirm the theoretical error estimates.

Abstract

The convergence of a family of AMF-Runge-Kutta methods (in short AMF-RK) for the time integration of evolutionary Partial Differential Equations (PDEs) of Advection Diffusion Reaction type semi-discretized in space is considered. The methods are based on very few inexact Newton Iterations of Aproximate Matrix Factorization splitting-type (AMF) applied to the Implicit Runge-Kutta formulas, which allows very cheap and inexact implementations of the underlying Runge-Kutta formula. Particular AMF-RK methods based on Radau IIA formulas are considered. These methods have given very competitive results when compared with important formulas in the literature for multidimensional systems of non-linear parabolic PDE problems. Uniform bounds for the global time-space errors on semi-linear PDEs when simultaneously the time step-size and the spatial grid resolution tend to zero are derived. Numerical illustrations supporting the theory are presented.