Stiff source terms inside ADER-DG method

ADER-DG numerical method with the LST-DG predictor, when used to solve hyperbolic equations with stiff source terms, exhibits properties characteristic of IMEX computational schemes. With increasing stiffness, characterized by the parameter $z=\nu {\mathrm{\Delta }t}^n<0$, the maximum of the Courant number ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ of the ADER-DG numerical method increases. In the region of strong stiffness $\mathrm{\mid }z\mathrm{\mid }\gg 1$, a linear increase in the Courant number ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(z)\sim \mathrm{\mid }z\mathrm{\mid }$ occurs. In the region of small source terms $\mathrm{\mid }z\mathrm{\mid }\ll 1$, the results ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(z)$ transform into known results for the problem ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(0)$ without source terms. In the transition region $\mathrm{\mid }z\mathrm{\mid }\sim 1$, a range of insignificant decrease in the Courant number ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(z)$ is discovered. In the linear case, instability is violated precisely when one of the matrix eigenvalues reaches $\lambda =-1$, regardless of the phase $\theta$ of the numerical solution. A rigorous stability theory of the ADER-DG numerical method for stiff problems for arbitrary degree $N$ of basis polynomials is constructed. The stability condition is simplified to solving a polynomial equation for the Courant number $\mathrm{C}\mathrm{F}\mathrm{L}$. Highly accurate dependencies of the Courant number ${\mathrm{C}\mathrm{F}\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(N,z)$ of the ADER-DG method on the parameter $z$ are obtained. A rigorous direct proof of the approximation of the ADER-DG method is presented. Approximation orders $p=N+1$ for arbitrary degrees $N$ are rigorously derived. A set of numerical experiments is carried out to apply the ADER-DG method to solving both a linear advection-reaction equation and the system of Euler equations for a two-component reacting medium. The results obtained in these calculations confirm the theoretical results well.