Stability of Newton TVD Runge-Kutta scheme for one-dimensional Euler equations with adaptive mesh

In this paper, we propose a moving mesh method with a Newton total variation diminishing (TVD) Runge-Kutta scheme for the Euler equations. Our scheme improves time discretization in the moving mesh algorithms. By analyzing the semi-discrete Euler equations with the discrete moving mesh equations as constraints, the stability of the Newton TVD Runge-Kutta scheme is proved. Thus, we can conclude that the proposed algorithm can generate a weak solution to the Euler equations. Finally, numerical examples are presented to verify the theoretical results and demonstrate the accuracy of the proposed scheme.