Well-balanced finite volume schemes for hydrodynamic equations with general free energy (1812.00980v3)

Abstract: Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases.

• The paper develops first- and second-order well-balanced finite volume schemes that preserve stationary states with machine precision and dissipate discrete free energy for various hydrodynamic systems.
• These schemes can be applied to a broad spectrum of physical problems, including phase transitions and collective dynamics, offering a unified framework derived from free energy.
• The research provides robust numerical methods validated across multiple test cases, enabling more accurate long-term simulations crucial for studying complex processes like interface movements and collective behaviors.

Overview of Well-Balanced Finite Volume Schemes for Hydrodynamic Equations with General Free Energy

The paper by Carrillo, Kalliadasis, Perez, and Shu focuses on developing well-balanced finite volume schemes with free energy dissipation for a broad class of hydrodynamic systems characterized by linear and nonlinear damping. These schemes are constructed to accurately preserve stationary states and dissipate free energy, crucial for studying the stability of these states in complex scenarios like phase transitions and collective behaviors.

Key Contributions

The authors propose first- and second-order accurate finite volume schemes that exhibit well-balanced properties and adhere to the dissipation of discrete free energy. The most critical feature of these schemes is their ability to preserve stationary states with machine precision, a significant challenge when using traditional numerical methods, such as fractional-step approaches, which introduce splitting errors and fail to maintain this balance at steady states.

A broad spectrum of physical problems can be evaluated using the proposed schemes, ranging from phase transitions described by density functional theory (DFT) to swarming dynamics modeled by the generalized Cucker-Smale model. The paper presents a meticulous formulation of these schemes, ensuring they handle various hydrodynamic systems under a common framework derived from the free energy functional.

Methodology

The authors address the persistence of steady-state relations and dissipative properties using the associated free energy. The hydrodynamic equations are cast into a general format incorporating density and momentum dynamics influenced by pressure, interaction potentials, and damping terms. A Lyapunov functional form based on free energy variation helps simplify the equations, leading to stationary state solutions that are preserved using well-balanced schemes.

The paper details a systematic way of treating source terms to maintain the well-balanced property and decay of free energy. This involves specific discretization choices for the potential terms, ensuring that interface values of the discrete solution satisfy these properties. The generalized potential term $H(x; \rho)$, incorporating both external fields and interaction potentials, is discretized to allow the accurate capture of stationary states and is consistent with familiar models like shallow water and hyperbolic chemotaxis equations.

Numerical Schemes and Validation

The constructed finite volume schemes, both first- and second-order, undergo rigorous validation using multiple test cases to confirm their well-balanced nature and accuracy. The paper showcases simulations demonstrating the schemes' ability to handle complex free energy forms, such as those arising in dynamic density functional theory (DDFT) applications.

The authors provide detailed numerical experiments and accuracy tests, ensuring their schemes accurately maintain stationary states in machine precision across different scenarios. These include fluid dynamics simulations with challenging interactions and damping factors, illustrating the robustness and applicability of the proposed methods.

Implications and Future Directions

The implications of this research are significant for both theoretical understanding and practical computation in hydrodynamics and related fields. Well-balanced schemes facilitate accurate long-term simulations of fluids involving intricate free energy dynamics, crucial for studying processes like phase transitions, interface movements, and collective behaviors in granular media.

The paper opens avenues for extending these methodologies to multi-dimensional settings, potentially applicable in analyzing surface diffusion processes relevant to materials science. The proposed framework could also serve in "data-driven" approaches for systems where experimental free energy descriptions are available, enabling a deeper exploration of bifurcations and phase transitions across varied physical contexts.

The authors highlight the potential to further develop computational methods that incorporate these well-balanced schemes for broader applications, suggesting potential extensions into computational studies involving multi-species systems or reactive mixtures where free energy plays a crucial role in system evolution.