High-Order Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations With Nonlocal Free Energy

Carrillo, José A.; Castro, Manuel J.; Kalliadasis, Serafim; Perez, Sergio P.

doi:10.1137/20m1332645

Cited by 7 publications

(4 citation statements)

References 80 publications

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“…In Fig. 2 we plot the temporal evolution of the 1D solution and the decay of the discrete free energy (17). For the latter we observe that the steady state is reached at t 0.1 as is evident from the plateau that the free energy at that time.…”

Section: Numerical Experimentsmentioning

confidence: 87%

“…Lastly, in Fig. 4(e) and (f) we illustrate the evolution of the discrete free energy (17), where we observe that the logarithmic potential takes slightly less time to reach the steady state.…”

Section: 21mentioning

confidence: 96%

“…Nevertheless, the extension of this scheme to high order will be explored in future works. The present study builds naturally from our previous works aimed at designing structure-preserving finite-volume schemes for gradient flows and hydrodynamic systems, where a general free energy containing nonlocal interaction potentials drives the temporal evolution towards a steady state dictated by the minimizer of such free energy [5,6,17,18,20,48].…”

Section: Substratementioning

confidence: 99%

See 2 more Smart Citations

Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation

Bailo¹,

Carrillo²,

Kalliadasis³

et al. 2021

Preprint

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We propose finite-volume schemes for the Cahn-Hilliard equation that unconditionally and discretely satisfy the boundedness of the phase field and the free-energy dissipation. Our numerical framework is applicable to a variety of free-energy potentials including the Ginzburg-Landau and Flory-Huggins, general wetting boundary conditions and degenerate mobilities. Its central thrust is the finite-volume upwind methodology, which we combine with a semi-implicit formulation based on the classical convex-splitting approach for the free-energy terms. Extension to an arbitrary number of dimensions is straightforward thanks to their cost-saving dimensional-splitting nature, which allows to efficiently solve higher-dimensional simulations with a simple parallelization. The numerical schemes are validated and tested in a variety of prototypical configurations with different numbers of dimensions and a rich variety of contact angles between droplets and substrates.

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Section: Numerical Experimentsmentioning

confidence: 87%

Section: 21mentioning

confidence: 96%

Section: Substratementioning

confidence: 99%

See 1 more Smart Citation

Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation

Bailo¹,

Carrillo²,

Kalliadasis³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recent work has considered stochastic DDFT (the Dean-Kawasaki equation) [177][178][179][180][181][182][183] and the McKean-Vlasov equation (a DDFT-type model) [184][185][186][187] from a mathematical perspective. Moreover, numerical methods were developed for DDFT [188][189][190][191][192][193][194][195][196] and PFC models [197][198][199][200]. DDFT was also used to test a new Brownian dynamics simulation method [201].…”

Section: Mathematics and Softwarementioning

confidence: 99%

Perspective: New directions in dynamical density functional theory

Vrugt

Wittkowski

2022

J. Phys.: Condens. Matter

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Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.

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Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics

et al. 2022

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We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection ‘flow’ vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in space and time variables, and utilizing variants of existing fixed-point methods as well as a recently developed Newton–Krylov scheme. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters, in both two and three dimensions. A key contribution is the provision of software which allows the discretization and solution of a range of optimization problems constrained by differential equations describing particle dynamics.

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High-Order Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations With Nonlocal Free Energy

Cited by 7 publications

References 80 publications

Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation

Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation

Perspective: New directions in dynamical density functional theory

Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics

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