Local DG Predictor in ADER-DG Schemes

Updated 27 January 2026

Local DG Predictor is a numerical scheme that employs local space–time DG solves to produce high-order polynomial approximations within each element.
It replaces classical Taylor expansions with a genuine local weak formulation to achieve enhanced stability, superconvergence, and subgrid resolution.
The method is central to modern ADER-DG frameworks and is applied in solving stiff ODEs, DAEs, and hyperbolic PDEs in reactive and complex flow applications.

A Local DG Predictor is an element-local discontinuous Galerkin (DG) solver applied in the temporal (and often spatial) dimension, producing a high-order-in-time (and -space) polynomial approximation of the solution within each element for the duration of a single time step. In the context of the ADER-DG (Arbitrary DERivative DG) framework, the local predictor replaces the classical Taylor expansion with a genuine local space–time DG solve, enabling arbitrarily high-order accuracy, subgrid time resolution, strong stability properties, and efficient handling of stiff source terms or reactions. These predictors are central to a variety of modern high-order finite element and finite volume methods for ODEs, DAEs, and hyperbolic PDE systems, including those with nonsmooth and stiff phenomena such as multidimensional reactive flow and detonation.

1. Mathematical Formulation

The canonical context is the solution of IVPs for systems of first-order ODEs or PDEs, either in pure time or in space–time cells. For a system

$\frac{d\mathbf u}{dt}=\mathbf F(\mathbf u,t),\quad \mathbf u(t_0)=\mathbf u_0,$

the time interval is decomposed into slabs $[t_n,t_{n+1}]$ , each locally mapped to a reference interval $\tau\in[0,1]$ via $t(\tau)=t_n+\Delta t_n\,\tau$ , with $\Delta t_n=t_{n+1}-t_n$ . The local solution in each cell is sought as a polynomial (or tensor polynomial for PDEs),

$\mathbf q_n(\tau) = \sum_{p=0}^N \hat{\mathbf q}_{n,p}\,\varphi_p(\tau),$

where $\varphi_p$ is a basis (typically Lagrange polynomials through Gauss–Legendre or Radau nodes). The predictor enforces the weak form

$\int_0^1\varphi_p(\tau) \left[ \frac{d\mathbf q_n}{d\tau} - \Delta t_n\, \mathbf F(\mathbf q_n(\tau), t(\tau)) \right] d\tau = 0, \quad \mathbf q_n(0)=\mathbf u_n,$

leading to a system of $(N+1)$ coupled nonlinear algebraic equations for the expansion coefficients, solved via Newton or fixed-point iteration. The same construction extends to the space–time context for PDEs, by introducing a tensor-product basis for each direction and constructing the weak form on the reference space–time cell (Popov, 2024, Popov, 19 Aug 2025).

2. Algorithmic Construction and Solution Features

The local predictor stage has the following general workflow (Popov, 2024, Popov, 19 Aug 2025):

Basis Setup: Precompute nodes and weights for quadrature (e.g., Gauss–Legendre or Radau points).
Initialization: Set the initial guess for all expansion coefficients to the value at the left node.
Nonlinear Iteration: In each iteration,
- Evaluate the right-hand side at quadrature points.
- Update coefficients using Newton or Picard methods, exploiting the structure and precomputed matrices (mass, stiffness).
Storage: The resulting polynomial is stored explicitly in nodal or modal form and can be used for accurate pointwise evaluation within the slab.

For PDEs, the procedure generalizes by adding multi-indexing over spatial bases and constructing Kronecker-product mass, stiffness, and flux matrices (Popov, 2024, Wolf et al., 2021).

For DAEs, the system is coupled for both differential and algebraic variables, and the local weak form enforces both components; Newton's method is applied to the full coupled system (Popov, 2024).

3. Accuracy, Stability, and Subgrid Resolution

The local DG predictor exhibits specific convergence and stability properties:

Order of convergence in the local polynomial: $N+1$ in $L^\infty$ and $L^2$ norms per time slab for polynomials of degree $N$ (Popov, 19 Aug 2025, Popov, 20 Jan 2026).
Superconvergence at grid nodes: The global ADER-DG update achieves order $2N+1$ at grid nodes (nodal superconvergence) (Popov, 2024, Popov, 19 Aug 2025, Popov, 20 Jan 2026).
Improved local continuity and order: The improved local solution is defined by

$\mathbf q_n^{\rm IL}(\tau) = \mathbf u_n + \Delta t_n\int_0^\tau \mathbf F(\mathbf q_n(\xi), t(\xi))\,d\xi,$

which is continuous at slab boundaries and attains order $N+2$ globally in $L^\infty$ and $L^2$ (Popov, 20 Jan 2026).

Stability: For linear test problems, the predictor inherits A-stability and L-stability (or $L_1$ -stability) properties. The stability function for Dahlquist's problem satisfies $|R(z)|<1$ for $\Re z<0$ and $|R(z)|\sim|z|^{-1}$ as $|z|\to\infty$ (Popov, 2024, Popov, 2024).
Subgrid Time Resolution: Evaluating the predictor polynomial at interior $\tau$ points yields highly accurate subgrid points, enabling reconstruction on coarse grids and detailed analysis of dynamics between nodes (Popov, 2024, Popov, 19 Aug 2025).

4. Application to Hyperbolic Conservation Laws and Reactive Flows

In high-order ADER-DG schemes for hyperbolic systems and reaction-convection equations, the local DG predictor is extended to the space–time element, with the local solution expanded as

$q_h(\xi, \tau) = \sum_{\mathfrak{p}} \hat{q}_{\mathfrak{p}} \Phi_{\mathfrak{p}}(\xi, \tau)$

using tensor-product Lagrange bases. The weak form integrates the time and spatial derivatives and relevant source terms, yielding a system for the expansion coefficients, solved locally per element per time step (Popov, 2024, Popov, 2024). The resulting high-order local solution is central to:

Non-splitting, high-order update: All volume and flux integrals needed for the explicit ADER-DG update are computed efficiently using the predictor polynomial.
A posteriori limiting: In troubled cells (detected via positivity and discrete maximum principle checks), the polynomial is replaced by a sub-cell finite volume WENO reconstruction, with the predictor re-run on sub-cells (Popov, 2024, Popov, 2024).
Adaptivity: The predictor naturally supports local time-stepping, mesh adaptation, and local time partitioning for cells with stiff reactions (Popov, 2024).

5. Computational Complexity and High-Performance Realization

The local DG predictor involves solving small (size $(N+1)\times d$ for ODEs, $(N+1)^D\times M$ for D-dimensional PDEs) nonlinear algebraic systems per element per step. Key properties (Popov, 2024, Wolf et al., 2021):

Iteration Count: Typically, 3–5 iterations for convergence in non-stiff problems; Newton (with Jacobian formation) is used for stiff sources.
Operation Count: For ODEs, $O((N+1)^2)$ right-hand side evaluations per step; for 3D PDEs, the number of unknowns grows as $Q(N+1)B(N)$ with $Q$ equations and $B(N)$ spatial basis functions.
Block-Structured Solvers: In high-performance implementations, the block upper-triangular structure is exploited with block-wise backsubstitution algorithms, and all tensor contractions are mapped to efficient small GEMMs via code generation tools, reducing the number of floating point operations by up to a factor of 25 at $N=6$ compared to dense LU (Wolf et al., 2021).
Scalability: The local nature allows maximal parallelism and efficient memory usage, as each element solve is independent.

6. Extensions, Modifications, and Theoretical Advances

Significant methodological advances and extensions include:

Higher regularity and continuity: The improved local solution construction ensures global-in-time $C^0$ continuity and an order-raising effect not present in the original predictor (Popov, 20 Jan 2026).
DAE Systems: An $\varepsilon$ -embedding and careful enforcement of algebraic constraints extend predictor applicability and convergence guarantees to index-1 DAE systems (Popov, 2024, Popov, 20 Jan 2026).
Space–Time Adaptive DG: The predictor is critical to adaptive ADER-DG methods employing both mesh refinement and asynchronous local time stepping (Popov, 2024, Wolf et al., 2021).
Regionally-Implicit DG Predictors: In RIDG methods, the predictor includes information from a small region of neighboring elements to remove the $O(1/p)$ strictness of the time step permitted by local predictors, increasing the stable CFL up to dimension-independent constants (Guthrey et al., 2017).
Subgrid Accuracy in Practice: Empirical results indicate that the predictor maintains near-nodal accuracy over a wide range of polynomial degrees and in the presence of extreme stiffness (Popov, 2024, Popov, 20 Jan 2026).

7. Representative Applications and Empirical Performance

Local DG predictors have enabled high-order robust simulation in diverse applications:

Compressible multicomponent and reacting flows: Used in the simulation of non-stationary compressible multicomponent reactive flows, including ZND detonation wave propagation without operator splitting (Popov, 2024, Popov, 2024).
Seismic waves in poroelastic media: Achieve high-order convergence, proper interface treatment, and efficient computation via block-structured solvers and code generation for clustered local time stepping (Wolf et al., 2021).
DAE and stiff ODE problems: Order $2N+1$ superconvergence and $N+1$ (or $N+2$ with improvements) local accuracy have been consistently verified across a wide range of degree $N$ and stiffness regimes (Popov, 19 Aug 2025, Popov, 20 Jan 2026, Popov, 2024).

Empirical order verification, scaling, and performance results consistently confirm the theoretical properties of the method, with strong evidence for the efficiency, accuracy, and flexibility of the local predictor approach.

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