On improving the efficiency of ADER methods

TL;DR

This work advances high-order ADER time integration by combining precise polynomial reconstructions, a Deferred Correction interpretation, and novel embedding strategies (ADERu, ADERdu, ADER–$L^2$) to achieve substantial efficiency gains. By exploiting GLB/GLG subtimenodes and linking ADER to implicit RK methods, the authors derive optimal iteration counts, enable $p$-adaptivity, and maintain stability. They also prove equivalences across bases and demonstrate that a natural DeC framing yields the minimal, order-consistent iterations needed for a given discretization, with robust linear stability properties. The approach is validated through extensive ODE and PDE tests, including SD-based hyperbolic PDEs, where the modified schemes achieve significant speed-ups while preserving accuracy and stability, especially in refined meshes. Overall, the paper provides a systematic pathway to faster, adaptive, high-order ADER schemes with practical PDE applications.

Abstract

The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient $p$-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization.

Figures (17)

• Figure 1: Sparsity pattern of the $A$ matrix with GLB subtimenodes and order 7. From left to right: cADER, ADER, ADERu and ADERdu (equivalent to ADER-$L^2$). The references to the stages indices are reported on the left and on top. In cADER the blocks have size 7, in ADER they have size 5, in ADERu and ADERdu they have increasing sizes from 2 to 5
• Figure 2: SD element for second and third order in $1$-dimension
• Figure 3: Linear system: Error decay for various methods and orders. The "ref" line is the reference for every order of accuracy
• Figure 4: Linear system: Error with respect to computational time.
• Figure 5: Linear system: Average number of iterations ($\pm$ half standard deviation) of adaptive ADERu (left) and ADERdu (right) for different time steps.

Theorems & Definitions (43)

• Remark 2.1: On the order of accuracy
• Remark 2.2: On the computation of $\boldsymbol{u\mkern-3mu}\mkern3mu_{n+1}$
• Remark 2.3: On the difference between ADER and ADER-IWF-RK and non-suitability for stiff problems
• Proposition 2.4
• Proposition 2.5: Convergence of the iterative procedure
• Theorem 2.6: Invertibility of $B$
• Theorem 3.1: The ADER-IWF \ref{['eq:weakproblemdiscrete']} is an implicit RK
• Proposition 3.2
• Proposition 3.3
• Theorem 3.4: Butcher 1964 butcher1964implicit