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.