DG Methods Using Dafermos' Entropy Rate Criterion for Unstructured Grids

Core Concepts

Dafermos' entropy rate criterion enhances stability in DG methods on unstructured grids.

Abstract

The article extends the approach to multidimensional systems of conservation laws using Discontinuous Galerkin methods on triangular grids. Special attention is given to entropy dissipation from boundaries, resulting in schemes free of tunable viscosity parameters. Theoretical background on hyperbolic systems and balance laws is discussed, along with the application of Dafermos' entropy rate criterion for numerical approximations. The implementation of filters for dissipating entropy and accuracy tests are also presented.

Introduction:

Solvers for hyperbolic systems are crucial in computational fluid dynamics.
Existence and uniqueness theory for associated equations remain open.

Theory:

Multidimensional DG schemes are high-order generalizations of Finite Volume methods.
Discontinuous Galerkin method allows discontinuities between cells.

Implementation:

Correction direction υ is calculated using a dissipation generator G.

Numerical Tests:

Accuracy test results show high-order accuracy and mesh independence for polynomial degrees 1 and 3.

Sedov Blast Wave:

Simulation of Sedov's blast wave problem shows non-symmetric solutions due to contact discontinuity instability.

Stats

Schemes satisfying additional criterion constructed [21, 22, 23].
Dissipation speed bounded by conservative filter [28].
Sedov blast wave initial conditions: ρin = 1.0, vin = 0, pin = 1.0, ρout = 0.125, vout = 0, pout = 0.1.

Quotes

Key Insights Distilled From

Stabilizing DG Methods Using Dafermos' Entropy Rate Criterion

by Simon-Christ... at arxiv.org 03-20-2024

https://arxiv.org/pdf/2403.12689.pdf

Deeper Inquiries

How does the application of Dafermos' entropy rate criterion impact the stability of DG methods

Dafermos' entropy rate criterion plays a crucial role in enhancing the stability of Discontinuous Galerkin (DG) methods. By enforcing this criterion, the numerical schemes are guided to dissipate entropy at a rate that ensures the preservation of physical properties and prevents spurious oscillations or instabilities. This leads to more robust and accurate simulations, especially for hyperbolic systems of conservation laws like the Euler equations in computational fluid dynamics. The application of Dafermos' entropy rate criterion helps maintain numerical stability by ensuring that the total entropy decreases over time as expected from physical principles.

What challenges may arise when implementing filters for dissipating entropy in numerical simulations

Implementing filters for dissipating entropy in numerical simulations can pose several challenges. One major challenge is designing filters that are both positive and conservative while effectively dissipating entropy without introducing artificial damping or altering the solution inaccurately. Ensuring that these filters do not introduce numerical artifacts or affect the overall accuracy of the simulation is essential but can be technically demanding. Additionally, determining appropriate filter strengths and incorporating them into existing DG methods without compromising their efficiency requires careful consideration.

How can the findings from the Sedov blast wave simulation be applied to real-world fluid dynamics problems

The findings from simulating Sedov's blast wave using unstructured triangular meshes provide valuable insights applicable to real-world fluid dynamics problems. Understanding how shock waves propagate through different mediums and interact with surrounding materials is crucial in various fields such as astrophysics, aerodynamics, and explosion modeling. By studying phenomena like blast waves numerically, researchers can gain deeper insights into shock physics, turbulence effects, and material interactions under extreme conditions. The ability to accurately model complex fluid dynamics scenarios like blast waves enhances our understanding of natural events and human-made processes involving high-energy releases or impact events.