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 $z = ν{Δt}^{n} < 0$, the maximum of the Courant number $\mathrm{CFL}_{\rm max}$ of the ADER-DG numerical method increases. In the region of strong stiffness $|z| \gg 1$, a linear increase in the Courant number $\mathrm{CFL}_{\rm max}(z) \sim |z|$ occurs. In the region of small source terms $|z| \ll 1$, the results $\mathrm{CFL}_{\rm max}(z)$ transform into known results for the problem $\mathrm{CFL}_{\rm max}(0)$ without source terms. In the transition region $|z| \sim 1$, a range of insignificant decrease in the Courant number $\mathrm{CFL}_{\rm max}(z)$ is discovered. In the linear case, instability is violated precisely when one of the matrix eigenvalues reaches $λ= -1$, regardless of the phase $θ$ 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 $\mathrm{CFL}$. Highly accurate dependencies of the Courant number $\mathrm{CFL}_{\rm max}(N, z)$ 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.