---
title: Local DG Predictor in ADER-DG Schemes
url: https://www.emergentmind.com/topics/local-dg-predictor
type: topic
---

# Local DG Predictor in ADER-DG Schemes

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 [2409.09932, 2508.13824].

## 2. Algorithmic Construction and Solution Features

The local predictor stage has the following general workflow [2409.09933, 2508.13824]:

- **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 [2409.09932, 2108.10565].

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 [2410.04018].

## 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 $N$ [2508.13824, 2601.13908].
- **Superconvergence at grid nodes**: The global ADER-DG update achieves order $2N+1$ at grid nodes (nodal superconvergence) [2409.09933, 2508.13824, 2601.13908].
- **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 $L^2$ [2601.13908].

- **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 $|z|\to\infty$ [2409.09933, 2410.04018].

- **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 [2409.09933, 2508.13824].

## 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 [2409.09932, 2409.09911]. 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 [2409.09932, 2409.09911].
- **Adaptivity**: The predictor naturally supports local time-stepping, mesh adaptation, and local time partitioning for cells with stiff reactions [2409.09911].

## 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 [2409.09933, 2108.10565]:

- **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 [2108.10565].
- **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 [2601.13908].
- **DAE Systems**: An $\varepsilon$-embedding and careful enforcement of algebraic constraints extend predictor applicability and convergence guarantees to index-1 DAE systems [2410.04018, 2601.13908].
- **Space–Time Adaptive DG**: The predictor is critical to adaptive ADER-DG methods employing both mesh refinement and asynchronous local time stepping [2409.09932, 2108.10565].
- **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 [1711.03447].
- **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 [2409.09933, 2601.13908].

## 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 [2409.09932, 2409.09911].
- **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 [2108.10565].
- **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 [2508.13824, 2601.13908, 2410.04018].

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.

---

References:  
- [2409.09932], [2409.09933], [2508.13824], [2108.10565], [2410.04018], [2409.09911], [2601.13908], [1711.03447].

Source: https://www.emergentmind.com/topics/local-dg-predictor