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Energy Stable Multigrid Method for Local and Non-local Hydrodynamic Models for Freezing
Numerical Analysis · Submitted 25 Mar 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Numerical simulations that confirm the stability and accuracy of the numerical method are presented.
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In this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing. The models take the form of compressible Navier-Stokes equations whose pressure is determined by the variational derivative of a free energy, which is a functional of the density field. We present unconditionally energy stable and mass conserving implicit finite difference methods for the models. The methods are based on a convex splitting of the free energy and that ensures that a discrete energy is non-increasing for any choice of time and space step. The methods are applicable to a large class of models, including both local and non-local free energy functionals. The theoretical basis for the numerical method is presented in a general context. The method is applied to problems using two specific free energy functionals: one local and one non-local functional. A nonlinear multigrid method is used to solve the numerical method, which is nonlinear at the implicit time step. The non-local functional, which is a convolution operator, is approximated using the Discrete Fourier Transform. Numerical simulations that confirm the stability and accuracy of the numerical method are presented.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing. | contribution 0.95 | 0.86 | 0.06 | asserted 1.00 |
| s1 | The models take the form of compressible Navier-Stokes equations whose pressure is determined by the variational derivative of a free energy, which is a functional of the density field. | background 0.97 | 0.16 | 0.04 | asserted 1.00 |
| s2 | We present unconditionally energy stable and mass conserving implicit finite difference methods for the models. | contribution 0.81 | 0.95 | 0.43 | asserted 1.00 |
| s3 | The methods are based on a convex splitting of the free energy and that ensures that a discrete energy is non-increasing for any choice of time and space step. | method 0.80 | 0.89 | 0.24 | asserted 1.00 |
| s4 | The methods are applicable to a large class of models, including both local and non-local free energy functionals. | contribution 0.84 | 0.66 | 0.10 | asserted 1.00 |
| s5 | The theoretical basis for the numerical method is presented in a general context. | contribution 0.74 | 0.76 | 0.04 | asserted 0.99 |
| s6 | The method is applied to problems using two specific free energy functionals: one local and one non-local functional. | method 0.57 | 0.93 | 0.08 | asserted 1.00 |
| s7 | A nonlinear multigrid method is used to solve the numerical method, which is nonlinear at the implicit time step. | method 1.00 | 0.87 | 0.06 | asserted 1.00 |
| s8 | The non-local functional, which is a convolution operator, is approximated using the Discrete Fourier Transform. | method 1.00 | 0.86 | 0.04 | asserted 1.00 |
| s9 | Numerical simulations that confirm the stability and accuracy of the numerical method are presented. | result 0.95 | 0.95 | 0.42 | asserted 1.00 |
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Original abstract
In this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing. The models take the form of compressible Navier-Stokes equations whose pressure is determined by the variational derivative of a free energy, which is a functional of the density field. We present unconditionally energy stable and mass conserving implicit finite difference methods for the models. The methods are based on a convex splitting of the free energy and that ensures that a discrete energy is non-increasing for any choice of time and space step. The methods are applicable to a large class of models, including both local and non-local free energy functionals. The theoretical basis for the numerical method is presented in a general context. The method is applied to problems using two specific free energy functionals: one local and one non-local functional. A nonlinear multigrid method is used to solve the numerical method, which is nonlinear at the implicit time step. The non-local functional, which is a convolution operator, is approximated using the Discrete Fourier Transform. Numerical simulations that confirm the stability and accuracy of the numerical method are presented.
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We note that in [ref] , the incompressible Navier-Stokes system was coupled with the PFC model.
A Navier Stokes Phase Field Crystal Model for Colloidal Suspensions · Introduction · 0.88
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