Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-Conservative Products

Core Concepts

The author presents an alternative finite difference WENO scheme for hyperbolic systems with non-conservative products, aiming to achieve exact conservation and high-order accuracy.

Abstract

The content discusses the development of efficient alternative finite difference Weighted Essentially Non-Oscillatory (WENO) schemes for hyperbolic systems with non-conservative products. It introduces the classical finite difference WENO method and the newer alternative finite difference WENO (AFD-WENO) method. The AFD-WENO algorithm is designed to handle hyperbolic systems with non-conservative products efficiently while maintaining high accuracy. The paper demonstrates that AFD-WENO outperforms FD-WENO in terms of speed and cost-effectiveness, especially in Peta- and Exascale computing scenarios. By transitioning to a precise conservation form when non-conservative products are absent, shock locations can be accurately captured by the method. The formulation presented allows flexibility in using different Riemann solvers and performs well on problems with stiff source terms. Various test problems are discussed to showcase the effectiveness of AFD-WENO for hyperbolic systems.

Stats

Higher order variants of AFD-WENO schemes don’t cost much more than lower order variants.
AFD-WENO outperforms FD-WENO at all orders.
The larger number of floating point operations associated with larger stencils are efficiently amortized by the CPU when designed to be cache friendly.

Quotes

Key Insights Distilled From

Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-Conservative Products

by Dinshaw S. B... at arxiv.org 03-05-2024

https://arxiv.org/pdf/2403.01266.pdf

Deeper Inquiries

How does the efficiency of AFD-WENO impact computational resources in practical applications

The efficiency of AFD-WENO has a significant impact on computational resources in practical applications. By offering high order accuracy at a fraction of the cost compared to finite volume WENO or DG schemes, AFD-WENO reduces the computational burden while maintaining precision. This efficiency translates into faster simulations and reduced resource requirements, making it particularly advantageous for large-scale problems or simulations that demand high levels of accuracy.

What challenges might arise when implementing AFD-WENO on complex multidimensional problems

Implementing AFD-WENO on complex multidimensional problems may pose several challenges. One challenge is ensuring stability and accuracy across multiple dimensions, as errors can propagate differently in higher-dimensional spaces. Additionally, handling non-conservative products in hyperbolic systems adds complexity to the algorithm implementation. Optimizing memory usage and computational speed while maintaining accuracy becomes crucial when dealing with intricate multidimensional scenarios.

How could the principles behind AFD-WENO be applied to other numerical methods or algorithms

The principles behind AFD-WENO can be applied to other numerical methods or algorithms by incorporating adaptive order interpolation techniques and flexible Riemann solvers. These concepts can enhance the performance of various numerical schemes by improving their adaptability to different problem domains and solution types. The idea of transitioning between conservation form and fluctuation form based on system characteristics can be generalized to develop versatile algorithms capable of efficiently handling diverse physical phenomena with varying degrees of complexity.