Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

Concetti Chiave

Developing bound-preserving CU schemes for hyperbolic conservation laws.

Sintesi

This article introduces a novel framework for analyzing and constructing bound-preserving (BP) central-upwind (CU) schemes for general hyperbolic systems of conservation laws. The authors address the challenges of preserving bounds in numerical schemes, focusing on the Euler equations of gas dynamics. By decomposing CU schemes into intermediate solution states, they simplify the design of BP CU schemes. The proposed approach is validated through numerical examples, demonstrating robustness and effectiveness.

Introduction:

Focus on high-order numerical schemes for hyperbolic systems.
Exact solutions satisfying bounds crucial in conservation laws.

Central-Upwind Schemes:

CU schemes reduce numerical dissipation compared to staggered central schemes.
Modifications to CU schemes improve accuracy and anti-diffusion properties.

Constructing BPCU Schemes:

Four essential conditions ensure bound preservation in CU schemes.
Convex decomposition simplifies analysis and modification steps.

1-D BPCU Schemes:

Review of 1-D CU scheme overview.
Proposal of BP framework and construction steps for 1-D Euler equations.

Numerical Examples:

Validation through demanding tests like shock diffraction problems.

Statistiche

None

Citazioni

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Approfondimenti chiave tratti da

Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws

by Shumo Cui,Al... alle arxiv.org 03-21-2024

https://arxiv.org/pdf/2403.13420.pdf

Domande più approfondite

How do high-order BP schemes compare to traditional methods

High-order BP schemes offer several advantages over traditional methods. Firstly, they provide higher accuracy and convergence rates, allowing for more precise numerical solutions. This is especially beneficial when dealing with complex systems of equations or problems that require a high level of accuracy. Additionally, high-order BP schemes are better at capturing sharp gradients and discontinuities in the solution, leading to more reliable results in regions of rapid change. Moreover, these schemes typically exhibit reduced numerical dissipation compared to lower-order methods, preserving important features of the solution such as shocks or rarefactions.

What are the limitations of applying GQL frameworks to other types of equations

While GQL frameworks have been successful in addressing bound-preserving (BP) problems for certain types of equations like hyperbolic conservation laws, there are limitations when applying them to other types of equations. One limitation is related to the complexity and nonlinearity of the invariant region defining the bounds. In cases where this region is highly nonlinear or implicit, it can be challenging to implement GQL techniques effectively. Additionally, some equations may not lend themselves well to convex decomposition approaches used in GQL frameworks due to their inherent properties or characteristics.

How can these findings be extended to multi-dimensional systems

The findings from the analysis and construction of BPCU schemes for one-dimensional systems can be extended to multi-dimensional systems by considering additional complexities introduced by spatial dimensions. For multi-dimensional systems, similar principles can be applied by decomposing the system into different directions and enforcing bound-preserving conditions independently along each dimension while ensuring consistency across all dimensions. The key lies in adapting the framework developed for one-dimensional systems to account for interactions between variables across multiple dimensions and maintaining stability and accuracy throughout the computational domain.