Improving the local solution of the DG predictor of the ADER-DG method for solving systems of ordinary differential equations and its applicability to systems of differential-algebraic equations

TL;DR

The paper tackles solving high-order ODE systems with the ADER-DG method by introducing an improved local solution that augments the standard local predictor. The key idea is to define q_n^{IL}(τ) via an integral form that preserves continuity at grid nodes and raises the local approximation order from $p_L = N+1$ to $p_{IL} = N+2$ without requiring major structural changes, while maintaining the original method's $p_G = 2N+1$ node convergence and stability. Rigorous analysis, including an ε-embedding framework, establishes these orders and extends applicability to DAE systems, with empirical results over polynomial degrees up to $N=60$ confirming the theory across Dahlquist, linear, and nonlinear test problems. The improved local solution enhances accuracy and smoothness in the inter-node domain, preserves node-based superconvergence, and broadens the ADER-DG method’s practical reach to DAEs, all while keeping implementation overhead minimal.

Abstract

Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence orders of one higher than the convergence order of the local numerical solution of the original ADER-DG numerical method and has the property of continuity at grid nodes. Rigorous proofs of the approximation orders of the local numerical solution and the improved local numerical solution are presented. Obtaining the proposed improved local numerical solution does not require significant changes to the structure of the ADER-DG numerical method. Therefore, all conclusions regarding the convergence orders of the numerical solution at grid nodes, the resulting superconvergence, and the high stability of the ADER-DG numerical method remain unchanged. A wide range of applications of the ADER-DG numerical method is presented for solving specific initial value problems for ODE systems for a wide range of polynomial degrees. The obtained results provide strong confirmation for the developed rigorous theory. The improved local numerical solution is shown to exhibit both higher accuracy and improved smoothness and point-wise comparability. Empirical convergence orders of all individual numerical solutions were calculated for a wide range of error norms, which well agree with the expected convergence orders. The rigorous proof, based on the $ε$-embedding method, of the applicability of the ADER-DG numerical method with a local DG predictor to solving DAE systems is presents.

Figures (7)

• Figure 1: Numerical solution of the problem (\ref{['eq:demo_ode']}), in the domain $0 \leqslant t \leqslant 8\pi$ with step $\mathrm{\Delta}t = \pi$. Comparison of the solution at nodes $\mathbf{u}_{n}$, the local solution $\mathbf{u}_{L}(t)$, the improved local solution $\mathbf{u}_{IL}(t)$ and the exact solution $\mathbf{u}^{\rm ex}(t)$ for components $u_{1} \equiv x$ (\ref{['fig:demo_nodes_8:a1']}, \ref{['fig:demo_nodes_8:a2']}, \ref{['fig:demo_nodes_8:b1']}, \ref{['fig:demo_nodes_8:b2']}, \ref{['fig:demo_nodes_8:c1']}, \ref{['fig:demo_nodes_8:c2']}) and $u_{2} \equiv \dot{x}$ (\ref{['fig:demo_nodes_8:a3']}, \ref{['fig:demo_nodes_8:a4']}, \ref{['fig:demo_nodes_8:b3']}, \ref{['fig:demo_nodes_8:b4']}, \ref{['fig:demo_nodes_8:c3']}, \ref{['fig:demo_nodes_8:c4']}), obtained using polynomials with degrees $N = 1$ (\ref{['fig:demo_nodes_8:a1']}, \ref{['fig:demo_nodes_8:a2']}, \ref{['fig:demo_nodes_8:a3']}, \ref{['fig:demo_nodes_8:a4']}), $N = 2$ (\ref{['fig:demo_nodes_8:b1']}, \ref{['fig:demo_nodes_8:b2']}, \ref{['fig:demo_nodes_8:b3']}, \ref{['fig:demo_nodes_8:b4']}) and $N = 3$ (\ref{['fig:demo_nodes_8:c1']}, \ref{['fig:demo_nodes_8:c2']}, \ref{['fig:demo_nodes_8:c3']}, \ref{['fig:demo_nodes_8:c4']}). Zoomed domain $0 \leqslant t \leqslant 10$ is presented on (\ref{['fig:demo_nodes_8:a2']}, \ref{['fig:demo_nodes_8:a4']}, \ref{['fig:demo_nodes_8:b2']}, \ref{['fig:demo_nodes_8:b4']}, \ref{['fig:demo_nodes_8:c2']}, \ref{['fig:demo_nodes_8:c4']}).
• Figure 2: Numerical solution of the problem (\ref{['eq:demo_ode']}), in the domain $0 \leqslant t \leqslant 8\pi$ with step $\mathrm{\Delta}t = 4\pi$. Comparison of the solution at nodes $\mathbf{u}_{n}$, the local solution $\mathbf{u}_{L}(t)$, the improved local solution $\mathbf{u}_{\rm IL}(t)$ and the exact solution $\mathbf{u}^{\rm ex}(t)$ for components $u_{1} \equiv x$ (\ref{['fig:demo_nodes_2:a1']}, \ref{['fig:demo_nodes_2:a2']}, \ref{['fig:demo_nodes_2:b1']}, \ref{['fig:demo_nodes_2:b2']}, \ref{['fig:demo_nodes_2:c1']}, \ref{['fig:demo_nodes_2:c2']}) and $u_{2} \equiv \dot{x}$ (\ref{['fig:demo_nodes_2:a3']}, \ref{['fig:demo_nodes_2:a4']}, \ref{['fig:demo_nodes_2:b3']}, \ref{['fig:demo_nodes_2:b4']}, \ref{['fig:demo_nodes_2:c3']}, \ref{['fig:demo_nodes_2:c4']}), obtained using polynomials with degrees $N = 4$ (\ref{['fig:demo_nodes_2:a1']}, \ref{['fig:demo_nodes_2:a2']}, \ref{['fig:demo_nodes_2:a3']}, \ref{['fig:demo_nodes_2:a4']}), $N = 8$ (\ref{['fig:demo_nodes_2:b1']}, \ref{['fig:demo_nodes_2:b2']}, \ref{['fig:demo_nodes_2:b3']}, \ref{['fig:demo_nodes_2:b4']}) and $N = 60$ (\ref{['fig:demo_nodes_2:c1']}, \ref{['fig:demo_nodes_2:c2']}, \ref{['fig:demo_nodes_2:c3']}, \ref{['fig:demo_nodes_2:c4']}). Zoomed domain $0 \leqslant t \leqslant 10$ is presented on (\ref{['fig:demo_nodes_2:a2']}, \ref{['fig:demo_nodes_2:a4']}, \ref{['fig:demo_nodes_2:b2']}, \ref{['fig:demo_nodes_2:b4']}, \ref{['fig:demo_nodes_2:c2']}, \ref{['fig:demo_nodes_2:c4']}).
• Figure 3: Numerical solution of the system (\ref{['eq:exp_diss_ode']}). Comparison of the solution at nodes $u_{n}$, the local solution $u_{L}(t)$, the improved local solution $u_{\rm IL}(t)$ and the exact solution $u^{\rm ex}(t)$ (\ref{['fig:exp_diss:a1']}, \ref{['fig:exp_diss:a2']}, \ref{['fig:exp_diss:a3']}, \ref{['fig:exp_diss:a4']}) and the errors $\varepsilon(t)$ (\ref{['fig:harm_osc:b1']}, \ref{['fig:harm_osc:b2']}, \ref{['fig:harm_osc:b3']}, \ref{['fig:harm_osc:b4']}), obtained using polynomials with degrees $N = 1$ (\ref{['fig:exp_diss:a1']}, \ref{['fig:exp_diss:b1']}), $N = 4$ (\ref{['fig:exp_diss:a2']}, \ref{['fig:exp_diss:b2']}), $N = 12$ (\ref{['fig:exp_diss:a3']}, \ref{['fig:exp_diss:b3']}) and $N = 60$ (\ref{['fig:exp_diss:a4']}, \ref{['fig:exp_diss:b4']}). Log-log plot of the dependence of the global error for the local solution $e^{l}$ (\ref{['fig:exp_diss:c1']}, \ref{['fig:exp_diss:c2']}, \ref{['fig:exp_diss:c3']}, \ref{['fig:exp_diss:c4']}), the improved local solution $e^{\rm imp}$ (\ref{['fig:exp_diss:d1']}, \ref{['fig:exp_diss:d2']}, \ref{['fig:exp_diss:d3']}, \ref{['fig:exp_diss:d4']}) and the solution at nodes $e^{n}$ (\ref{['fig:exp_diss:e1']}, \ref{['fig:exp_diss:e2']}, \ref{['fig:exp_diss:e3']}, \ref{['fig:exp_diss:e4']}) on the discretization step $\mathrm{\Delta}t$, obtained in the $f$-norm and norms $L_{1}$, $L_{2}$ and $L_{\infty}$, obtained using polynomials with degrees $N = 1$ (\ref{['fig:exp_diss:c1']}, \ref{['fig:exp_diss:d1']}, \ref{['fig:exp_diss:e1']}), $N = 4$ (\ref{['fig:exp_diss:c2']}, \ref{['fig:exp_diss:d2']}, \ref{['fig:exp_diss:e2']}), $N = 12$ (\ref{['fig:exp_diss:c3']}, \ref{['fig:exp_diss:d3']}, \ref{['fig:exp_diss:e3']}) and $N = 60$ (\ref{['fig:exp_diss:c4']}, \ref{['fig:exp_diss:d4']}, \ref{['fig:exp_diss:e4']}).
• Figure 4: Numerical solution of the system (\ref{['eq:lin_diss_ode']}). Comparison of the solution at nodes $\mathbf{u}_{n}$, the local solution $\mathbf{u}_{L}(t)$, the improved local solution $\mathbf{u}_{\rm IL}(t)$ and the exact solution $\mathbf{u}^{\rm ex}(t)$ (\ref{['eq:lin_diss_sol_ex']}) for components $u_{1} \equiv x$ (\ref{['fig:lin_diss:a1']}, \ref{['fig:lin_diss:a2']}, \ref{['fig:lin_diss:a3']}, \ref{['fig:lin_diss:a4']}) and $u_{2} \equiv \dot{x}$ (\ref{['fig:lin_diss:b1']}, \ref{['fig:lin_diss:b2']}, \ref{['fig:lin_diss:b3']}, \ref{['fig:lin_diss:b4']}), the errors $\varepsilon(t)$ (\ref{['fig:lin_diss:c1']}, \ref{['fig:lin_diss:c2']}, \ref{['fig:lin_diss:c3']}, \ref{['fig:lin_diss:c4']}), obtained using polynomials with degrees $N = 1$ (\ref{['fig:lin_diss:a1']}, \ref{['fig:lin_diss:b1']}, \ref{['fig:lin_diss:c1']}), $N = 4$ (\ref{['fig:lin_diss:a2']}, \ref{['fig:lin_diss:b2']}, \ref{['fig:lin_diss:c2']}), $N = 12$ (\ref{['fig:lin_diss:a3']}, \ref{['fig:lin_diss:b3']}, \ref{['fig:lin_diss:c3']}) and $N = 60$ (\ref{['fig:lin_diss:a4']}, \ref{['fig:lin_diss:b4']}, \ref{['fig:lin_diss:c4']}). Log-log plot of the dependence of the global error for the local solution $e^{l}$ (\ref{['fig:lin_diss:d1']}, \ref{['fig:lin_diss:d2']}, \ref{['fig:lin_diss:d3']}, \ref{['fig:lin_diss:d4']}), the improved local solution $e^{\rm imp}$ (\ref{['fig:lin_diss:e1']}, \ref{['fig:lin_diss:e2']}, \ref{['fig:lin_diss:e3']}, \ref{['fig:lin_diss:e4']}) and the solution at nodes $e^{n}$ (\ref{['fig:lin_diss:f1']}, \ref{['fig:lin_diss:f2']}, \ref{['fig:lin_diss:f3']}, \ref{['fig:lin_diss:f4']}) on the discretization step $\mathrm{\Delta}t$, obtained in the $f$-norm and norms $L_{1}$, $L_{2}$ and $L_{\infty}$, obtained using polynomials with degrees $N = 1$ (\ref{['fig:lin_diss:d1']}, \ref{['fig:lin_diss:e1']}, \ref{['fig:lin_diss:f1']}), $N = 4$ (\ref{['fig:lin_diss:d2']}, \ref{['fig:lin_diss:e2']}, \ref{['fig:lin_diss:f2']}), $N = 12$ (\ref{['fig:lin_diss:d3']}, \ref{['fig:lin_diss:e3']}, \ref{['fig:lin_diss:f3']}) and $N = 60$ (\ref{['fig:lin_diss:d4']}, \ref{['fig:lin_diss:e4']}, \ref{['fig:lin_diss:f4']}).
• Figure 5: Numerical solution of the system (\ref{['eq:harm_osc_ode']}). Comparison of the solution at nodes $\mathbf{u}_{n}$, the local solution $\mathbf{u}_{L}(t)$, the improved local solution $\mathbf{u}_{\rm IL}(t)$ and the exact solution $\mathbf{u}^{\rm ex}(t)$ (\ref{['eq:harm_osc_sol_ex']}) for components $u_{1} \equiv x$ (\ref{['fig:harm_osc:a1']}, \ref{['fig:harm_osc:a2']}, \ref{['fig:harm_osc:a3']}, \ref{['fig:harm_osc:a4']}) and $u_{2} \equiv \dot{x}$ (\ref{['fig:harm_osc:b1']}, \ref{['fig:harm_osc:b2']}, \ref{['fig:harm_osc:b3']}, \ref{['fig:harm_osc:b4']}), the errors $\varepsilon(t)$ (\ref{['fig:harm_osc:c1']}, \ref{['fig:harm_osc:c2']}, \ref{['fig:harm_osc:c3']}, \ref{['fig:harm_osc:c4']}), obtained using polynomials with degrees $N = 1$ (\ref{['fig:harm_osc:a1']}, \ref{['fig:harm_osc:b1']}, \ref{['fig:harm_osc:c1']}), $N = 4$ (\ref{['fig:harm_osc:a2']}, \ref{['fig:harm_osc:b2']}, \ref{['fig:harm_osc:c2']}), $N = 12$ (\ref{['fig:harm_osc:a3']}, \ref{['fig:harm_osc:b3']}, \ref{['fig:harm_osc:c3']}) and $N = 60$ (\ref{['fig:harm_osc:a4']}, \ref{['fig:harm_osc:b4']}, \ref{['fig:harm_osc:c4']}). Log-log plot of the dependence of the global error for the local solution $e^{l}$ (\ref{['fig:harm_osc:d1']}, \ref{['fig:harm_osc:d2']}, \ref{['fig:harm_osc:d3']}, \ref{['fig:harm_osc:d4']}), the improved local solution $e^{\rm imp}$ (\ref{['fig:harm_osc:e1']}, \ref{['fig:harm_osc:e2']}, \ref{['fig:harm_osc:e3']}, \ref{['fig:harm_osc:e4']}) and the solution at nodes $e^{n}$ (\ref{['fig:harm_osc:f1']}, \ref{['fig:harm_osc:f2']}, \ref{['fig:harm_osc:f3']}, \ref{['fig:harm_osc:f4']}) on the discretization step $\mathrm{\Delta}t$, obtained in the $f$-norm and norms $L_{1}$, $L_{2}$ and $L_{\infty}$, obtained using polynomials with degrees $N = 1$ (\ref{['fig:harm_osc:d1']}, \ref{['fig:harm_osc:e1']}, \ref{['fig:harm_osc:f1']}), $N = 4$ (\ref{['fig:harm_osc:d2']}, \ref{['fig:harm_osc:e2']}, \ref{['fig:harm_osc:f2']}), $N = 12$ (\ref{['fig:harm_osc:d3']}, \ref{['fig:harm_osc:e3']}, \ref{['fig:harm_osc:f3']}) and $N = 60$ (\ref{['fig:harm_osc:d4']}, \ref{['fig:harm_osc:e4']}, \ref{['fig:harm_osc:f4']}).