The effective use of BLAS interface for implementation of finite-element ADER-DG and finite-volume ADER-WENO methods

TL;DR

This paper presents an efficient method for implementing ADER-DG and ADER-WENO numerical methods using BLAS interface and JIT technologies, significantly improving computational performance in high-performance computing environments.

Contribution

It introduces a novel approach integrating BLAS with JIT to optimize ADER family methods, reducing computational costs and simplifying implementation complexity.

Findings

Implementation with BLAS and JIT outperforms traditional methods by several orders of magnitude.

The approach operates efficiently on AoS data structures without transposition.

Performance analysis explains the cost reduction through roofline analysis.

Abstract

Numerical methods of the ADER family, in particular finite-element ADER-DG and finite-volume ADER-WENO methods, are among the most accurate numerical methods for solving quasilinear hyperbolic PDE systems. The internal structure of ADER-DG and ADER-WENO numerical methods contains a large number of basic linear algebra operations related to matrix multiplications. The main interface of software libraries for matrix multiplications for high-performance computing is BLAS. An effective method for integration the standard functions of the BLAS interface into the implementation of these numerical methods is presented. The calculated matrices are small matrices; and this allows to use effectively JIT technologies. The proposed approach immediately operates on AoS, which allows to efficiently calculate flux, source and non-conservative terms without transposition. The obtained computational costs demonstrated that the effective implementation, based on the use of the JIT functions of the BLAS, outperformed both the implementation based on the general BLAS functions and the vanilla implementations by several orders of magnitude. The complexity of developing an implementation based on the proposed approach does not exceed the complexity of developing a vanilla implementation. Performance analysis using roofline partly explains the observed features of the decreasing of computational costs.