Robustness of High-Order Upwind Summation-By-Parts Methods for Nonlinear Conservation Laws

Belangrijkste concepten

High-order upwind SBP methods ensure robustness and stability for nonlinear conservation laws.

Samenvatting

The article discusses the application of upwind SBP operators to nonlinear conservation laws. It explores different flux vector splittings, local stability analysis, and numerical testing on various meshes. The formulation of upwind SBP methods for challenging flow examples is a key focus.

Statistieken

Mattsson (2017) developed upwind SBP operators.
Gassner, Svärd, and Hindenlang (2022) studied local stability.
Various flux vector splittings are analyzed.

Citaten

"We investigate the robustness of upwind SBP methods for challenging examples of shock-free flows."

Belangrijkste Inzichten Gedestilleerd Uit

On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws

by Hend... om arxiv.org 03-28-2024

https://arxiv.org/pdf/2311.13888.pdf

Diepere vragen

How do upwind SBP methods compare to other numerical schemes for conservation laws

Upwind SBP methods offer several advantages compared to other numerical schemes for conservation laws. One key advantage is their ability to provide provably stable and robust solutions for hyperbolic conservation laws. By incorporating artificial dissipation through the upwind operators, these methods can handle shock waves and discontinuities effectively without sacrificing accuracy. Additionally, upwind SBP methods are well-suited for under-resolved flows, where traditional high-order methods may struggle due to stability issues. The use of flux vector splitting allows for a more flexible and tailored approach to capturing the dynamics of the system accurately.

What are the limitations of using high-order methods for under-resolved flows

While high-order methods are known for their accuracy and efficiency, they come with limitations when applied to under-resolved flows. One major limitation is the potential for numerical instabilities, especially in the presence of shocks or discontinuities. In such cases, high-order methods may exhibit oscillations or spurious solutions, leading to inaccurate results. Additionally, the computational cost of high-order methods can be prohibitive for under-resolved flows, where resolving all the details of the flow may not be necessary or feasible. Tuning the parameters of high-order methods for under-resolved flows can also be challenging and may require extensive computational resources.

How can the findings of this study be applied to real-world fluid dynamics problems

The findings of this study on the robustness of high-order upwind SBP methods for nonlinear conservation laws have significant implications for real-world fluid dynamics problems. By demonstrating the effectiveness of these methods in handling challenging examples such as shock-free flows of the compressible Euler equations, like the Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex, researchers and practitioners in the field of fluid dynamics can apply these methods to improve the accuracy and stability of their simulations. The ability of upwind SBP methods to provide built-in dissipation and stability properties makes them well-suited for a wide range of applications, including atmospheric flows, shallow water equations, and scalar conservation laws. By incorporating different flux vector splittings and analyzing their impact on robustness, researchers can tailor these methods to specific fluid dynamics problems and achieve more reliable and accurate results.