Arbitrarily High-Order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models

We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models

Fully Discrete Second-Order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities

We develop spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel energy quadratization technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then shown to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second-order accuracy of the schemes. The linearization technique developed in this paper is so general that it can be applied to any thermodynamically consistent hydrodynamic theories so long as their energies are bounded below. Numerical examples on coarsening dynamics of two immiscible fluids and a heavy fluid drop settling in a lighter fluid matrix are presented to show the effectiveness of the proposed linear schemes. Predictions by the two fluid mixture models are compared and discussed, leading to our conclusion that the quasiincompressible model is more reliable than the incompressible one

Fully Discrete Second-Order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities

We develop spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel energy quadratization technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then shown to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second-order accuracy of the schemes. The linearization technique developed in this paper is so general that it can be applied to any thermodynamically consistent hydrodynamic theories so long as their energies are bounded below. Numerical examples on coarsening dynamics of two immiscible fluids and a heavy fluid drop settling in a lighter fluid matrix are presented to show the effectiveness of the proposed linear schemes. Predictions by the two fluid mixture models are compared and discussed, leading to our conclusion that the quasiincompressible model is more reliable than the incompressible one

On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics

This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge–Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations.Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit

Second Order Fully Discrete Energy Stable Methods on Staggered Grids for Hydrodynamic Phase Field Models of Binary Viscous Fluids

We present second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. We apply the energy quadratization strategy to develop a linear-implicit scheme. We then extend it to a decoupled, linear scheme by introducing an intermediate velocity term so that the phase variable, velocity field, and pressure can be solved sequentially. The two new, fully discrete linear schemes are then shown to be unconditionally energy stable, and the linear systems resulting from the schemes are proved uniquely solvable. Rates of convergence of the two linear schemes in both space and time are verified numerically. The decoupled scheme tends to introduce excessive dissipation compared to the coupled one. The coupled scheme is then used to simulate fluid drops of one fluid in the matrix of another fluid as well as mixing dynamics of binary polymeric, viscous solutions. The numerical results in mixing dynamics reveals the dramatic difference between the morphology in the simulations obtained using the two different boundary conditions (physical vs. periodic), demonstrating the importance of using proper boundary conditions in fluid dynamics simulations

Numerical solution of the complex Ginzburg-Landau equation alternatives for its time integration

Treball de Fi de Màster Universitari en Matemàtica Computacional (Pla de 2013). Codi: SIQ527. Curs 2021/2022 (A distància)La ecuación Ginzburg-Landau compleja en su forma cúbica describe una amplia gama de fenómenos para muchos sistemas de la física y presenta muchas estructuras coherentes estables entre sus soluciones. De ahí que se haya estudiado intensamente a lo largo de los años y que sea un laboratorio ideal para aprender sobre los esquemas de integración temporal. Desde el punto de vista numérico se pueden utilizar métodos pseudoespectrales para resolver la discretización espacial, hay que resolverlo con variable compleja y se pueden utilizar métodos específicos para la integración temporal como la Exponential Time Differencing o Integrating Factor Methods gracias a la presencia de términos lineales y no lineales integrables. El objetivo de esta tesis es comparar algunos de estos métodos analizando su orden y eficiencia. Para ello, se han programado los diferentes esquemas en Fortran y se han validado sus resultados mediante una pila de casos de prueba. A continuación, utilizando uno de ellos como referencia y calculando la solución de una prueba con un paso de tiempo muy pequeño, se han comparado los demás. Entre otros resultados, se ha constatado la buena convergencia de los métodos de orden superior o la mejor velocidad de los métodos que integran analíticamente parte de la ecuación o que necesitan menos transformadas de Fourier