[Paper Review] Efficient Implementation of ADER Schemes for Euler and Magnetohydrodynamical Flows on Structured Meshes -- Comparison with Runge-Kutta Methods

Dinshaw S. Balsara, Chad Meyer+3 more|arXiv (Cornell University)|Jun 10, 2010

Computational Fluid Dynamics and Aerodynamics47 references5 citations

TL;DR

This paper presents a practical, efficient implementation of ADER (Arbitrary DERivative) schemes for solving Euler and magnetohydrodynamical (MHD) equations on structured meshes, offering a single-step time integration method that outperforms traditional strong stability preserving Runge-Kutta schemes. The authors provide detailed formulations in both nodal and modal spaces, optimize numerical flux and electric field calculations for divergence-free MHD, and demonstrate that ADER schemes are nearly twice as fast as Runge-Kutta methods across all tested orders of accuracy.

ABSTRACT

ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update can be carried out in a single step, which is desirable in many applications. However, prior papers have focused on the theory while downplaying implementation details. The purpose of the present paper is to make ADER schemes accessible by providing two useful formulations of the method as well as their implementation details on three-dimensional structured meshes. We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related detail. We also provide details for the efficient use of ADER schemes in obtaining the numerical flux for conservation laws as well as electric fields for divergence-free magnetohydrodynamics. An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. The schemes catalogued here have been implemented in the first author's RIEMANN code. The speed of ADER schemes is shown to be almost twice as fast as that of strong stability preserving Runge-Kutta time stepping schemes for all the orders of accuracy that we tested.

Motivation & Objective

• To bridge the gap between theoretical ADER schemes and practical implementation in computational fluid dynamics.
• To provide detailed, usable formulations of ADER methods in nodal and modal spaces for structured meshes.
• To optimize numerical flux and electric field computation for divergence-free MHD simulations.
• To present an efficient WENO-based strategy for zone-averaged magnetic fields from face-centered data.
• To demonstrate the computational superiority of ADER over strong stability preserving Runge-Kutta schemes in terms of speed and accuracy.

Proposed method

• ADER schemes are implemented using a single-step time update, avoiding the stage-by-stage progression of Runge-Kutta methods.
• The method employs nodal and modal formulations for solving hyperbolic conservation laws with non-stiff source terms on 3D structured meshes.
• Numerical fluxes are computed efficiently using high-order accurate Riemann solvers tailored for conservation laws.
• Electric fields in MHD are computed directly from the flux integral to maintain divergence-free magnetic fields.
• A WENO-based reconstruction strategy is used to compute zone-averaged magnetic fields from face-centered magnetic field data.
• All schemes are implemented and validated in the RIEMANN code, enabling high-order accuracy and performance evaluation.

Experimental results

Research questions

• RQ1 How can ADER schemes be efficiently implemented in practice for structured meshes in computational fluid dynamics?
• RQ2 What are the key implementation details that enable high performance in ADER schemes for Euler and MHD equations?
• RQ3 How does the computational cost of ADER compare to strong stability preserving Runge-Kutta schemes across different orders of accuracy?
• RQ4 What strategies ensure divergence-free magnetic fields in ADER-based MHD simulations?
• RQ5 Can ADER schemes achieve near-doubling of computational speed relative to Runge-Kutta methods while maintaining high-order accuracy?

Key findings

• ADER schemes achieve nearly twice the computational speed of strong stability preserving Runge-Kutta schemes across all tested orders of accuracy.
• The implementation of ADER in nodal and modal spaces enables efficient, high-order accurate solutions on structured meshes.
• The use of a WENO-based reconstruction strategy ensures accurate and stable zone-averaged magnetic fields from face-centered data.
• Efficient computation of numerical fluxes and electric fields is critical for maintaining accuracy and divergence-free constraints in MHD.
• The ADER method's single-step time integration provides a significant performance advantage over multi-stage Runge-Kutta methods.
• The RIEMANN code implementation confirms the robustness and scalability of the proposed ADER formulations for complex fluid dynamics problems.