ADER-DG Numerical Method

• ADER-DG is a high-order, explicit one-step discontinuous Galerkin method designed to solve hyperbolic and hyperbolic-parabolic PDEs with adaptive mesh refinement.
• The method employs a local space–time predictor combined with a MOOD subcell finite-volume limiter to maintain accuracy and stability near discontinuities.
• Its efficiency is enhanced by compact stencils, optimized tensor contractions, and parallel scalability, making it well-suited for complex, multi-physics simulations.

The ADER-DG (Arbitrary high-order DERivative Discontinuous Galerkin) numerical method is a high-order, fully discrete, explicit one-step discontinuous Galerkin scheme for the numerical solution of hyperbolic and hyperbolic-parabolic partial differential equations (PDEs). The scheme is based on a combination of a local space–time predictor for attaining high-order in both space and time, an a posteriori subcell finite-volume limiting for non-oscillatory shock capturing, and an adaptive mesh refinement (AMR) framework with local time-stepping. ADER-DG has been successfully applied to special relativistic and general-relativistic magnetohydrodynamics (MHD), compressible Navier-Stokes, resistive MHD, dispersive and reactive flows, and numerous other nonlinear PDE systems with complex physics (Zanotti et al., 2015).

1. High-Order Space–Time Formulation and Predictor Step

The primary distinguishing feature of ADER-DG is its use of a single high-order (in both space and time) update per element and per time step. Given a conservation law

$\partial_t u + \nabla \cdot f(u) = 0,$

the DG solution is represented on each cell $T_i$ by a polynomial expansion of degree $N$: $$u_h(x, t^n)|_{T_i} = \sum_{\ell=0}^N \Phi_\ell(x) \hat{u}_\ell^n.$$ A local space–time predictor $q_h(x, t)$ is computed for each cell, which is a polynomial in both space and time and satisfies the governing equations in a weak form over the space–time slab $T_i \times [t^n, t^{n+1}]$ (Zanotti et al., 2015, Fambri et al., 2016, Fambri et al., 2018): $$\int_{T_E} \theta_k(\cdot, 1) \theta_\ell(\cdot, 1) d\xi \; q_\ell - \int_0^1 \int_{T_E} (\partial_\tau \theta_k) \theta_\ell d\xi d\tau \; q_\ell = \int_{T_E} \theta_k(\cdot, 0) \Phi_\ell d\xi \; \hat{u}_\ell^n - \int_0^1 \int_{T_E} \theta_k \nabla_\xi \theta_\ell d\xi d\tau \; f^*(q_\ell).$$ This local nonlinear system is solved in each cell independently (e.g., by Picard iteration). The local predictor enables space–time quadrature of all volume and surface integrals required for high-order accuracy.

2. One-Step DG Update and Riemann Solvers

The global update for each element is performed as a single step, avoiding multi-stage Runge-Kutta procedures: $$\left[\int_{T_i} \Phi_k \Phi_\ell dx\right](\hat{u}_\ell^{n+1} - \hat{u}_\ell^n) + \int_{t^n}^{t^{n+1}} \int_{\partial T_i} \Phi_k G(q_h^-, q_h^+) \cdot n dS dt - \int_{t^n}^{t^{n+1}} \int_{T_i} \nabla\Phi_k \cdot f(q_h) dx dt = 0,$$ where $G$ is a consistent numerical flux, such as Rusanov or HLL, possibly extended for non-conservative products with DLM path-conservative jump terms (Fambri et al., 2018, Bassi et al., 2020). Nodal or modal bases (typically Lagrange polynomials through Gauss–Legendre points) are used, and the element coupling is limited to nearest neighbors.

3. Subcell Finite-Volume Limiting (MOOD Paradigm)

While high-order DG is accurate in smooth regions, it can yield spurious oscillations near discontinuities. ADER-DG supplements the unlimited scheme with an a posteriori subcell finite-volume (FV) limiter based on the MOOD paradigm (Fambri et al., 2016):

• After each time step, the candidate DG solution $u_h^*$ is computed.
• Detection criteria are applied in each cell:
  • Physical checks: successful primitive recovery, positivity ($\rho, p > 0$), velocities bounded by causality ($|v| < c$).
  • Numerical checks: the value must satisfy a relaxed discrete maximum principle (DMP) in the cell and its neighbors.
• If either criterion fails, the cell is flagged as “troubled.”
• For troubled cells, the DG polynomial is projected onto subcell averages on a subgrid ($N_s=2N+1$ per direction).
• These subcell averages are advanced by a robust FV scheme (e.g., ADER-WENO or TVD).
• The high-order DG polynomial is reconstructed from the evolved subcell averages, ensuring consistency.
• Cells not flagged are updated by the unlimited DG step, maintaining high-order accuracy and efficiency (Zanotti et al., 2015).

This limiting strategy preserves the subcell resolution and robustness at shocks/discontinuities while maintaining high-order accuracy elsewhere.

4. Adaptive Mesh Refinement and Local Time-Stepping

ADER-DG is natively compatible with dynamic, cell-by-cell adaptive mesh refinement (Zanotti et al., 2015, Fambri et al., 2016, Fambri et al., 2018, Krenz et al., 2019):

• Cells are refined or coarsened based on smoothness indicators, such as second derivatives or total variation criteria.
• Each cell maintains a status (active, virtual child, or virtual parent) and exchanges data with neighbors at most one refinement level different.
• Local time-stepping is implemented at each refinement level, with $\Delta t_\ell = \Delta t_0 / r^\ell$, reducing computational cost in spatially adaptive contexts.
• All limiter and FV procedures are carried out consistently across AMR boundaries, with data transfer handled via $L_2$ projection (refinement) or averaging (coarsening).
• Local FV updates on subcells are fully compatible with AMR, and troubled cells are protected from coarsening until (numerical) admissibility is restored.

The algorithm proceeds backward in time through AMR levels, advancing the finest first, with fluxes conserved at level boundaries.

5. Accuracy, Efficiency, and Parallel Scalability

ADER-DG achieves arbitrary order $(N+1)$ in space and time in smooth regions and robust first-order convergence in the presence of discontinuities, provided by the subcell MOOD limiter (Zanotti et al., 2015, Fambri et al., 2016, Fambri et al., 2018). The explicit one-step time integration requires only a single communication (face data exchange) per time step, optimizing data locality and making the scheme highly suitable for hybrid shared/distributed-memory architectures (Dumbser et al., 2018, Krenz et al., 2019).

Key efficiency features:

• Only face neighbors are accessed per update (compact stencil).
• The predictor step is performed locally and can be vectorized and cache-blocked.
• All main integrals are reduced to small-tensor contractions, for which optimized BLAS Level-3 routines can be employed (Popov, 2024).
• Parallel scalability is demonstrated up to $16,000$ cores (80% efficiency) (Dumbser et al., 2018), with the ADER-DG scheme outperforming classical RKDG and FV schemes in wall-clock time per degree of freedom and communication cost.

6. Benchmark Applications and Theoretical Guarantees

ADER-DG with subcell MOOD limiting and AMR has been verified in:

• Special and general relativistic MHD, including Riemann and Kelvin–Helmholtz instability tests, Orszag–Tang vortex, and magnetized blast wave problems, resolving fine-scale, turbulent, and discontinuous structures (Zanotti et al., 2015, Fambri et al., 2018).
• Compressible Navier-Stokes and resistive MHD, where the scheme captures sharp fronts and boundary layers while preserving stability (Fambri et al., 2016).
• Dispersive and reactive flows, shock-dominated flows, detonation problems, and free-surface-seismic coupling (Bassi et al., 2020, Popov, 2024).

Empirical and theoretical studies confirm:

• Order $(N+1)$ convergence in smooth settings up to order $N=9$.
• First-order convergence at discontinuities, with limiter active only in a small fraction of cells.
• Robust stability in the presence of strong shocks, stiff source terms, or dissipative physics.

The scheme is widely applicable to astrophysical, geophysical, and engineering problems requiring efficiency, accuracy, and robustness on complex AMR meshes and modern heterogeneous high-performance computing platforms (Zanotti et al., 2015, Fambri et al., 2016).

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References

• "Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement" (Zanotti et al., 2015)
• "Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations" (Fambri et al., 2016)
• "ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics" (Fambri et al., 2018)
• "Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine" (Dumbser et al., 2018)
• "Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation" (Popov, 2024)
• "A High-Order Discontinuous Galerkin Solver with Dynamic Adaptive Mesh Refinement to Simulate Cloud Formation Processes" (Krenz et al., 2019)
• "The effective use of BLAS interface for implementation of finite-element ADER-DG and finite-volume ADER-WENO methods" (Popov, 2024)