Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations

TL;DR

This work adapts the arbitrary high order ADER-DG method with a local DG predictor to initial-value problems for first-order ODE systems, delivering a fully one-step solver with uniform steps formulated via domain partition and a reference-domain transform. It establishes $A$- and $L$-stability and demonstrates superconvergence for node solutions with order $p_G=2N+1$, while the local predictor yields $p_L=N+1$, with the local solution enabling subgrid resolution on coarse grids. The approach is validated on six classical IVP tests and stiff problems, showing accurate high-order performance and robust handling of stiffness, albeit with potential order degradation under extreme stiffness. Computational-cost discussions compare ADER-DG with standard Runge-Kutta methods, highlighting the trade-off between higher achievable order and per-step work, and underscoring the practical benefit of subgrid resolution via the local solution. The results indicate a scalable, high-order, stable framework for solving nonlinear ODE systems with potential advantages for stiff dynamics and coarse-gridded simulations.

Abstract

An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform steps, and is simple in algorithmic and software implementations. It was shown that the proposed version of the ADER-DG numerical method is A-stable and L-stable. The ADER-DG numerical method demonstrates superconvergence with convergence order 2N+1 for the solution at grid nodes, while the local solution obtained using the local DG predictor has convergence order N+1. It was demonstrated that an important applied feature of this implementation of the numerical method is the possibility of using the local solution as a solution with a subgrid resolution, which makes it possible to obtain a detailed solution even on very coarse coordinate grids. The scale of the error of the local solution, when calculating using standard representations of single or double precision floating point numbers, using large values of the degree N, practically does not differ from the error of the solution at the grid nodes. The capabilities of the ADER-DG method for solving stiff ODE systems characterized by extreme stiffness are demonstrated. Estimates of the computational costs of the ADER-DG numerical method are obtained.

Figures (12)

• Figure 1: Regions of absolute stability $|R(z)| < 1$ (outer part of closed curves) in the complex plane $z$ of the ADER-DG numerical method with a local DG predictor for the degrees $1 \leqslant N \leqslant 60$ of polynomials in the DG representation: $N = 1,\, 2,\, 3,\, 4,\, 5$ (a), $N = 6,\, 7,\, 8,\, 9$ (b), $N = 10,\, 15,\, 20,\, 25$ (c), $N = 30,\, 40,\, 50,\, 60$ (d). Horizontal and vertical axes represent real $\Re(z)$ and imaginary $\Im(z)$ parts of complex number $z = \lambda\cdot\mathrm{\Delta}t_{n}$.
• Figure 2: The absolute values of the stability function $|R(z)|$ of the ADER-DG numerical method with a local DG predictor for the degrees $N$ on a set of radial rays $z = |z|\exp(i\,\arg(z))$ in the complex plane $z$, with $0.5\pi \leqslant \arg(z) < 2.5\pi$: $N = 1$ (a), $N = 8$ (b), $N = 20$ (c), $N = 40$ (d). The line $|R(z)| \sim |z|^{-1}$ demonstrates the asymptotic behavior appropriate in the $L$-stability. The line $|R(z)| = 1$ shows the region of absolute stability; the rays $\arg(z) = 1.75\pi,\, 2.0\pi,\, 2.25\pi$ have unstable regions with $|R(z)| > 1$ corresponding to the regions inside the closed curves in Fig. \ref{['fig:stab_domain']}.
• Figure 3: Numerical solution of the problem (\ref{['eq:harm_osc_prec']}), in the domain $0 \leqslant t \leqslant 100\pi$ with step $\mathrm{\Delta}t = \frac{100}{96}\pi$, using the ADER-DG numerical method with a local DG predictor with degrees of polynomials $N = 8$ (\ref{['fig:harm_osc_sols_prec:a1']}, \ref{['fig:harm_osc_sols_prec:a2']}, \ref{['fig:harm_osc_sols_prec:a3']}), $16$ (\ref{['fig:harm_osc_sols_prec:b1']}, \ref{['fig:harm_osc_sols_prec:b2']}, \ref{['fig:harm_osc_sols_prec:b3']}) and $32$ (\ref{['fig:harm_osc_sols_prec:c1']}, \ref{['fig:harm_osc_sols_prec:c2']}, \ref{['fig:harm_osc_sols_prec:c3']}). Presented data: numerical solution at nodes, local solution and exact solution separately for components $u_{1}$ (\ref{['fig:harm_osc_sols_prec:a1']}, \ref{['fig:harm_osc_sols_prec:b1']}, \ref{['fig:harm_osc_sols_prec:c1']}) and $u_{2}$ (\ref{['fig:harm_osc_sols_prec:a2']}, \ref{['fig:harm_osc_sols_prec:b2']}, \ref{['fig:harm_osc_sols_prec:c2']}); dependence of the point-wise error $\varepsilon(t)$ of the numerical solution for the solution at the nodes and the local solution (\ref{['fig:harm_osc_sols_prec:a3']}, \ref{['fig:harm_osc_sols_prec:b3']}, \ref{['fig:harm_osc_sols_prec:c3']}).
• Figure 4: Numerical solution of the problem (\ref{['eq:harm_osc']}). Comparison of the solution at nodes $\mathbf{u}_{n}$, the local solution $\mathbf{u}_{L}(t)$ and the exact solution $\mathbf{u}^{\rm ex}(t)$ for components $u_{1}$ (\ref{['fig:harm_osc_sols:a1']}, \ref{['fig:harm_osc_sols:b1']}, \ref{['fig:harm_osc_sols:c1']}) and $u_{2}$ (\ref{['fig:harm_osc_sols:a2']}, \ref{['fig:harm_osc_sols:b2']}, \ref{['fig:harm_osc_sols:c2']}), the error $\varepsilon(t)$ (\ref{['fig:harm_osc_sols:a3']}, \ref{['fig:harm_osc_sols:b3']}, \ref{['fig:harm_osc_sols:c3']}), obtained using polynomials with degrees $N = 1$ (\ref{['fig:harm_osc_sols:a1']}, \ref{['fig:harm_osc_sols:a2']}, \ref{['fig:harm_osc_sols:a3']}), $N = 8$ (\ref{['fig:harm_osc_sols:b1']}, \ref{['fig:harm_osc_sols:b2']}, \ref{['fig:harm_osc_sols:b3']}) and $N = 60$ (\ref{['fig:harm_osc_sols:c1']}, \ref{['fig:harm_osc_sols:c2']}, \ref{['fig:harm_osc_sols:c3']}). Dependence of the global error for the solution at nodes $e_{G}$ (\ref{['fig:harm_osc_sols:d1']}, \ref{['fig:harm_osc_sols:d2']}, \ref{['fig:harm_osc_sols:d3']}) and the local solution $e_{L}$ (\ref{['fig:harm_osc_sols:e1']}, \ref{['fig:harm_osc_sols:e2']}, \ref{['fig:harm_osc_sols:e3']}) on the discretization step $\mathrm{\Delta}t$, obtained in the norms $L_{1}$, $L_{2}$ and $L_{\infty}$, obtained using polynomials with degrees $N = 1$ (\ref{['fig:harm_osc_sols:d1']}, \ref{['fig:harm_osc_sols:e1']}), $N = 8$ (\ref{['fig:harm_osc_sols:d2']}, \ref{['fig:harm_osc_sols:e2']}) and $N = 60$ (\ref{['fig:harm_osc_sols:d3']}, \ref{['fig:harm_osc_sols:e3']}).
• Figure 5: Numerical solution of the problem (\ref{['eq:exp_diss']}). Comparison of the solution at nodes $\mathbf{u}_{n}$, the local solution $\mathbf{u}_{L}(t)$ and the exact solution $\mathbf{u}^{\rm ex}(t)$ for components $u_{1}$ (\ref{['fig:exp_diss_sols:a1']}, \ref{['fig:exp_diss_sols:b1']}, \ref{['fig:exp_diss_sols:c1']}) and $u_{2}$ (\ref{['fig:exp_diss_sols:a2']}, \ref{['fig:exp_diss_sols:b2']}, \ref{['fig:exp_diss_sols:c2']}), the error $\varepsilon(t)$ (\ref{['fig:exp_diss_sols:a3']}, \ref{['fig:exp_diss_sols:b3']}, \ref{['fig:exp_diss_sols:c3']}), obtained using polynomials with degrees $N = 1$ (\ref{['fig:exp_diss_sols:a1']}, \ref{['fig:exp_diss_sols:a2']}, \ref{['fig:exp_diss_sols:a3']}), $N = 8$ (\ref{['fig:exp_diss_sols:b1']}, \ref{['fig:exp_diss_sols:b2']}, \ref{['fig:exp_diss_sols:b3']}) and $N = 60$ (\ref{['fig:exp_diss_sols:c1']}, \ref{['fig:exp_diss_sols:c2']}, \ref{['fig:exp_diss_sols:c3']}). Dependence of the global error for the solution at nodes $e_{G}$ (\ref{['fig:exp_diss_sols:d1']}, \ref{['fig:exp_diss_sols:d2']}, \ref{['fig:exp_diss_sols:d3']}) and the local solution $e_{L}$ (\ref{['fig:exp_diss_sols:e1']}, \ref{['fig:exp_diss_sols:e2']}, \ref{['fig:exp_diss_sols:e3']}) on the discretization step $\mathrm{\Delta}t$, obtained in the norms $L_{1}$, $L_{2}$ and $L_{\infty}$, obtained using polynomials with degrees $N = 1$ (\ref{['fig:exp_diss_sols:d1']}, \ref{['fig:exp_diss_sols:e1']}), $N = 8$ (\ref{['fig:exp_diss_sols:d2']}, \ref{['fig:exp_diss_sols:e2']}) and $N = 60$ (\ref{['fig:exp_diss_sols:d3']}, \ref{['fig:exp_diss_sols:e3']}).