Stiff source terms inside ADER-DG method
Abstract
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 = νΔtn < 0, the maximum of the Courant number CFLmax of the ADER-DG numerical method increases. In the region of strong stiffness |z| ≫ 1, a linear increase in the Courant number CFLmax(z) ∼ |z| occurs. In the region of small source terms |z| ≪ 1, the results CFLmax(z) transform into known results for the problem CFLmax(0) without source terms. In the transition region |z| ∼ 1, a range of insignificant decrease in the Courant number CFLmax(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 CFL. Highly accurate dependencies of the Courant number CFLmax(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.
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BibTeX
@misc{popov2026stiff,
title = {{Stiff source terms inside ADER-DG method}},
author = {I. S. Popov and A. P. Popova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30852},
eprint = {2610.30852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30852},
}
RIS
TY - UNPB
TI - Stiff source terms inside ADER-DG method
AU - I. S. Popov
AU - A. P. Popova
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30852
UR - https://arxiv.org/abs/2610.30852
AB - 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 = νΔt^n < 0, the maximum of the Courant number CFL_max of the ADER-DG numerical method increases. In the region of strong stiffness |z| ≫ 1, a linear increase in the Courant number CFL_max(z) ∼ |z| occurs. In the region of small source terms |z| ≪ 1, the results CFL_max(z) transform into known results for the problem CFL_max(0) without source terms. In the transition region |z| ∼ 1, a range of insignificant decrease in the Courant number CFL_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 CFL. Highly accurate dependencies of the Courant number CFL_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.
ER -