Application of Stabilized Explicit Runge-Kutta Methods to the Incompressible Navier-Stokes Equations by means of a Projection Method and a Differential Algebraic Approach

TL;DR

This thesis compares second-order stabilized explicit Runge-Kutta methods for incompressible Navier-Stokes equations, finding ROCK2 most efficient and versatile, while the differential algebraic approach offers better accuracy and stability despite higher computational cost.

Contribution

It provides a comparative analysis of RKC, ROCK2, and PIROCK schemes with projection and differential algebraic approaches for Navier-Stokes equations, highlighting their stability, accuracy, and efficiency.

Findings

ROCK2 is the most efficient and versatile method.

PIROCK exhibited unexpected instabilities.

Differential algebraic approach offers better accuracy and stability.

Abstract

In this master thesis we have compared different second order stabilized explicit Runge-Kutta methods when applied to the incompressible Navier-Stokes equations by means of a projection method and a differential algebraic approach. We explored the stability and accuracy properties of the RKC, ROCK2 and PIROCK schemes when coupled with the projection and the differential algebraic approach. PIROCK has shown unexpected instabilities, ROCK2 resulted to be the most efficient and versatile Runge-Kutta method taken into account. The differential algebraic approach sounds computationally costly but it exhibits better accuracy and a larger stability region. These properties make it more efficient than the projection method. The theory presented in the first chapters is supported by numerical experiments.