On improving the efficiency of ADER methods

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.

References in corpus (15)

• Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free Magnetohydrodynamics
• Very High Order $\PNM$ Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
• Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting
• High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
• Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
• Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
• High order ADER schemes for continuum mechanics
• High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics
• A multi-level spectral deferred correction method
• Interweaving PFASST and Parallel Multigrid
• Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes
• Asymptotic preserving Deferred Correction Residual Distribution schemes
• An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations
• Multi-Level Spectral Deferred Corrections Scheme for the Shallow Water Equations on the Rotating Sphere
• Spectral Difference method with a posteriori limiting: Application to the Euler equations in one and two space dimensions

Cited by in corpus (2)

• 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
• Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations