Energy Stable Method Research Articles

A novel diffuse-interface model and a fully-discrete maximum-principle-preserving energy-stable method for two-phase flow with surface tension and non-matching densities

Two well-established classes of the interface capturing models are the level-set and phase-field models. Level-set formulations satisfy the maximum principle for the density but are not energy-stable. On the other hand, phase-field models do satisfy the second law of thermodynamics but lack the maximum principle for the density. In this paper we derive a novel model for incompressible immiscible two-phase flow with non-matching densities and surface tension that is both energetically-stable and satisfies the maximum principle for the density. The model finds its place at the intersection of level-set and phase-field models. Its derivation is based on a diffusification of the incompressible two-phase Navier–Stokes equations with non-matching densities and surface tension and involves functional entropy variables. Additionally, we present an associated fully-discrete energy-stable method. Isogeometric analysis is used for the spatial discretization and the temporal-integration is performed with a new time-integration scheme that is a perturbation of the second-order midpoint scheme. The fully-discrete scheme is unconditionally energy-dissipative, pointwise divergence-free and satisfies the maximum principle for the density. Numerical examples in two and three dimensions verify the energetic-stability of the methodology.

Asymptotic Behaviour of Time Stepping Methods for Phase Field Models

Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error sigma in the limit of small length scale parameter epsilon rightarrow 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity.

Efficient and energy stable method for the Cahn-Hilliard phase-field model for diblock copolymers

In this paper, we consider numerical approximations to solving the Cahn-Hilliard phase field model for diblock copolymers. We combine the recently developed SAV (scalar auxiliary variable) approach with the stabilization technique to arrive at a novel stabilized-SAV method, where a crucial linear stabilization term is added to enhancing the stability and keeping the required accuracy while using the large time steps. The scheme is very easy-to-implement and fast in the sense that one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, convex-splitting, and non-stabilized SAV scheme for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically.

Energy Stability of Explicit Runge--Kutta Methods for Nonautonomous or Nonlinear Problems

Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge--Kutta methods for nonlinear or nonautonomous problems. We provide both necessary and sufficient conditions for energy stability over these classes of problems. Examples of conditionally energy stable schemes are constructed, and an example is given in which unconditional energy stability is obtained with an explicit scheme.

An energy stable method for the Swift–Hohenberg equation with quadratic–cubic nonlinearity

We present temporally first- and second-order accurate methods for the Swift–Hohenberg (SH) equation with quadratic–cubic nonlinearity. In order to handle the nonconvex, nonconcave term in the energy for the SH equation, we add an auxiliary term to make the combined term convex, which yields a convex–concave decomposition of the energy. As a result, the first- and second-order methods are unconditionally uniquely solvable and unconditionally stable with respect to the energy and pseudoenergy of the SH equation, respectively. And the Fourier spectral method is used for the spatial discretization. We present numerical examples showing the accuracy and energy stability of the proposed methods and the effect of the quadratic term in the SH equation on pattern formation.

Linear energy stable methods for an epitaxial growth model with slope selection

We consider a phase field model for molecular beam epitaxial growth with slope selection with the goal of determining linear energy stable time integration methods for the dynamics. Stable methods for this model have been found via a concave-convex splitting of the dynamics, but this approach generally leads to a nonlinear update equation. We seek a linear energy stable method to allow for simple and efficient time marching with fast Fourier transforms. Our approach is to parametrize a class of semi-implicit methods and perform unconditional von Neumann stability analysis to identify the region of stability in parameter space. Since unconditional von Neumann stability does not ensure energy stability, we perform extensive numerical tests and find strong agreement between the predicted and observed stable regions of parameter space. This analysis elucidates a novel feature that the stability region in parameter space differs for a mono-domain system (single equilibrium slope) versus a many-domain system (coarsening facets from an initially flat surface). The utility of these steps is then demonstrated by a comparison of the coarsening dynamics for isotropic and anisotropic variants of the model.

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier–Stokes–Cahn–Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman–Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with non-solenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and breakup dynamics of droplets including pinching due to gravity.

An energy-stable method for solving the incompressible Navier–Stokes equations with non-slip boundary condition

We introduce a stable method for solving the incompressible Navier–Stokes equations with variable density and viscosity. Our method is stable in the sense that it does not increase the total energy of dynamics that is the sum of kinetic energy and potential energy. Instead of velocity, a new state variable is taken so that the kinetic energy is formulated by the L2 norm of the new variable. Navier–Stokes equations are rephrased with respect to the new variable, and a stable time discretization for the rephrased equations is presented.Taking into consideration the incompressibility in the Marker-And-Cell (MAC) grid, we present a modified Lax–Friedrich method that is L2 stable. Utilizing the discrete integration-by-parts in MAC grid and the modified Lax–Friedrich method, the time discretization is fully discretized. An explicit CFL condition for the stability of the full discretization is given and mathematically proved.

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.

Energy stable and high-order-accurate finite difference methods on staggered grids

For wave propagation over distances of many wavelengths, high-order finite difference methods on staggered grids are widely used due to their excellent dispersion properties. However, the enforcement of boundary conditions in a stable manner and treatment of interface problems with discontinuous coefficients usually pose many challenges. In this work, we construct a provably stable and high-order-accurate finite difference method on staggered grids that can be applied to a broad class of boundary and interface problems. The staggered grid difference operators are in summation-by-parts form and when combined with a weak enforcement of the boundary conditions, lead to an energy stable method on multiblock grids. The general applicability of the method is demonstrated by simulating an explosive acoustic source, generating waves reflecting against a free surface and material discontinuity.

First- and second-order energy stable methods for the modified phase field crystal equation

The phase field crystal (PFC) model was extended to the modified phase field crystal (MPFC) model, which is a sixth-order nonlinear damped wave equation, to include not only diffusive dynamics but also elastic interactions. In this paper, we present temporally first- and second-order accurate methods for the MPFC equation, which are based on an appropriate splitting of the energy for the PFC equation. And we use the Fourier spectral method for the spatial discretization. The first- and second-order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the MPFC equation, respectively. Numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods.

Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted $L^2$ inner product. In applications where the wavespeed varies spatially at a subelement scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method that reduces storage costs by replacing the weighted $L^2$ inner product with a weight-adjusted inner product. This equivalent inner product results in an energy-stable method but does not increase storage costs for locally varying weights. A priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.

An unconditionally energy-stable method for the phase field crystal equation

The phase field crystal equation has been recently put forward as a model for microstructure evolution of two-phase systems on atomic length and diffusive time scales. The theory is cast in terms of an evolutive nonlinear sixth-order partial differential equation for the interatomic density that locally minimizes an energy functional with the constraint of mass conservation. Here we propose a new numerical algorithm for the phase field crystal equation that is second-order time-accurate and unconditionally stable with respect to the energy functional. We present several numerical examples in two and three dimensions dealing with crystal growth in a supercooled liquid and crack propagation in a ductile material. These examples show the effectiveness of our new algorithm.

Use of Energy-State Analysis on a Generic Air-Breathing Hypersonic Vehicle

Apossiblenext-generation launch vehiclemaybea fullyreusable,single-stage-to-orbitaerospacecraft.Thisclass of hypersonic vehicle includes highly integrated airframe and propulsion systems; hence fuel-optimal trajectories are of special interest. Also, because these vehicles are neither conventional aircraft nor rockets, the appropriate approach for trajectory analysis during preliminary design, for example, is of interest. Addressed here is whether the energy-state method is justie ed for the determination of such trajectories for scramjet-powered, hypersonic vehicles. The focus is on the scramjet-powered phase of the mission. The energy-state approach is known to be justie ed when the system dynamics exhibit multi-time-scale behavior, and such behavior is shown to exist for the vehicle and mission phase under investigation here. Furthermore, solutions obtained via the energy-state assumption are compared with a full dynamic solution obtained using a nonlinear programming routine. It is shown that the results from the two methods are in reasonably good agreement. On the basis of these results, the energy-state method would appear justie ed for the class of vehicle in question, at least for obtaining rapid performance estimates, as well as gaining insight into the basic performance issues.

Fixed-Range Optimum Trajectories for Short-Haul Aircraft

An algorithm, based on the energy-state method, is derived for calculating optimum trajectories with a range constraint. The basis of the algorithm is the assumption that optimum trajectories consist of, at most, three segments: an increasing energy segment (climb); a constant energy segment (cruise); and a decreasing energy segment (descent). This assumption allows energy to be used as the independent variable in the increasing and decreasing energy segments, thereby eliminating the integration of a separate adjoint differential equation and simplifying the calculus of variations problem to one requiring only pointwise extremization of algebraic functions. The algorithm is used to compute minimum fuel, minimum time, and minimum direct-operating-cost trajectories, with range as a parameter, for an in-service CTOL aircraft and for an advanced STOL aircraft. For the CTOL aircraft and the minimum-fuel performance function, the optimum controls, consisting of air-speed and engine power setting, are continuous functions of the energy in both climb and descent as well as near the maximum or cruise energy. This is also true for the STOL aircraft except in the descent where at one energy level a nearly constant energy dive segment occurs, yielding a discontinuity in the airspeed at that energy. The reason for this segment appears to be the relatively high fuel flow at idle power of the engines used by this STOL aircraft. Use of a simplified trajectory which eliminates the dive increases the fuel consumption of the total descent trajectory by about 10 percent and the time to fly the descent by about 19 percent compared to the optimum.