There is a large literature of numerical methods for phase field models from materials science. The prototype models are the Allen-Cahn and Cahn-Hilliard equations. We present four benchmark problems for these equations, with numerical results validated using several computational methods with different spatial and temporal discretizations. Our goal is to provide the scientific community with a reliable reference point for assessing the accuracy and reliability of future software for this important class of problem.

With the observation that the TENO weighting strategy can explicitly distinguish smooth scales from nonsmooth scales in spectral space, in this paper, a new discontinuity indicator is proposed based on the high-order TENO paradigm [Fu et al., JCP 305(2016): 333-359]. The local flow structures are classified as smooth or nonsmooth scales, and the hybrid numerical discretization scheme is applied correspondingly, i.e. the high-order upwind linear scheme without characteristic decomposition is employed for resolving smooth scales while the nonlinear low-dissipation TENO scheme is adopted to capture discontinuities. Since the time-consuming characteristic decomposition and smoothness-indicator computation of TENO are avoided in smooth regions, the overall computational efficiency can be improved significantly. Moreover, the cut-off wavenumber separating smooth and nonsmooth scales is determined by the parameter C_T. In contrast to the thresholds of other discontinuity indicators, which are typically defined in physical space, C_T takes effects in wavespace rendering its high generality. A set of benchmark cases with widespread length-scales is simulated to assess the performance of the proposed discontinuity indicator and the resulting hybrid shock-capturing scheme. Compared to the monotonicity-preserving discontinuity indicator and the TVB discontinuity indicator, the proposed algorithm delivers better performance with a fixed set of parameters for all considered benchmarks.

Recently, a new family of splitting methods, the so-called vector penalty-projection methods (VPP) were introduced by Angot et al. [4, 7] to compute the solution of unsteady incompressible fluid flows and to overcome most of the drawbacks of the usual incremental projection methods. Two different parameters are related to the VPP methods: the augmentation parameter r ≥0 and the penalty parameter 0<ε≤1. In this paper, we deal with the time-dependent incompressible Stokes equations with open boundary conditions using the VPP methods. The spatial discretization is based on the finite volume scheme on a Marker and Cells (MAC) staggered grid. Furthermore, two different second-order time discretization schemes are investigated: the second-order Backward Difference Formula (BDF2) also known as Gear's scheme and the Crank-Nicolson scheme. We show that the VPP methods provide a second-order convergence rate for both velocity and pressure in space and time even in the presence of open boundary conditions with small values of the augmentation parameter r typically 0≤r≤1 and a penalty parameter ε small enough typically ε=10⁻¹⁰. The resulting constraint on the discrete divergence of velocity is not exactly equal to zero but is satisfied approximately as O(εδt) where ε is the penalty parameter (taken as small as desired) and δt is the time step. The choice r=0 requires special attention to avoid the accumulation of the round-off errors for very small values of ε. Indeed, it is important in this case to directly correct the pressure gradient by taking account of the velocity correction issued from the vector penalty-projection step. Finally, the efficiency and the second-order accuracy of the method are illustrated by several numerical test cases including homogeneous or non-homogeneous given traction on the boundary.

This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k−1, and k (k ≥1) for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees l, k, l (l=k−1, k) for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number.

In this paper we consider numerical solutions of the diffuse interface model with Peng-Robinson equation of state for the multi-component two-phase fluid system, which describes real states of hydrocarbon fluids in petroleum industry. A major challenge is to develop appropriate temporal discretizations to overcome the strong nonlinearity of the source term and preserve the energy dissipation law in the discrete sense. Efficient first and second order time stepping schemes are designed based on the "Invariant Energy Quadratization" approach and the stabilized method. The resulting temporal semi-discretizations by both schemes lead to linear systems that are symmetric and positive definite at each time step, and their unconditional energy stabilities are rigorously proven. Numerical experiments are presented to demonstrate accuracy and stability of the proposed schemes.

In this paper, a mass conservative lattice Boltzmann model (LBM) is proposed to simulate the two-phase flows with moving contact lines at high density ratio. The proposed model consists of a phase field lattice Boltzmann equation (LBE) for solving the conservative Allen-Cahn (A-C) equation, and a pressure evolution LBE for solving the incompressible Navier-Stokes equations. In addition, a modified wall boundary treatment scheme is developed to ensure the mass conservation. The wetting dynamics are treated by incorporating the cubic wall energy in the expression of the total free energy. The current model is characterized by mass conservation, proper treatment of wetting boundary and high density ratio. We applied the model on a series of numerical tests including equilibrium droplets on wetting surfaces, co-current flow and a droplet moving by gravity along inclined wetting surfaces. Theoretical analysis and experiments were conducted for model validation. The numerical results show good performances on mass conservation even with a density contrast up to 1000. Furthermore, the results show that the moving contact line can be successfully recovered, which proves that this model is applicable on the study of moving contact line issue and further related applications.

In this paper, a new nonlinear finite volume scheme preserving positivity for three-dimensional (3D) heat conduction equation is proposed. Being different from the traditional monotone schemes, the flux on each 3D non-planar cell-face is entirely approximated by the so-called effective directional flux firstly, then the effective directional flux is decomposed by the fixed stencils. Fixed stencil means the decomposition is just conducted on this face such that searching the convex decomposition stencil over all cell-faces is avoided. This feature makes our scheme more efficient than the traditional monotone ones based on the adaptive stencils for convex decompositions, especially in 3D. In addition, similar to other schemes based on the fixed stencils, there is also no assumption of the non-negativity of the interpolated cell-vertex unknowns. Some benchmark examples are presented to demonstrate the second-order accuracy. Two anisotropic diffusion problems show that not only can our schemes maintain the positivity-preserving property, but also they are more efficient than the scheme based on the adaptive stencil.

We propose a simple numerical algorithm to estimate the finite time Lyapunov exponent (FTLE) in dynamical systems from only a sparse number of Lagrangian particle trajectories. The method first reconstructs the flow field using the radial basis function (RBF) and then uses either the Lagrangian or the Eulerian approach to determine the corresponding flow map. We also develop a simple algorithm based on the Schur complement for updating, rather than recomputing, the reconstruction in the RBF when new trajectory data is made available in applications. We will demonstrate the effectiveness of the proposed method using examples from autonomous and aperiodic flows, and also measurements from real-life data.

A full approximation storage multigrid method has been shown to improve the convergence properties of the D2Q9 BGK lattice Boltzmann method [14]. In this paper we extend the method to the multiple relaxation time collision model due to its improved stability. The new method is validated using Poiseuille flow. The method is then extended to three dimensions. The efficiency of the V-cycle and W-cycle multigrid schedules and the under-relaxed Jacobi and Gauss Seidel multigrid smoothers are reported for the two and three dimensional models for lid driven cavity flow.

The projector augmented-wave (PAW) method is an important approach for electronic structure calculations based on the Kohn–Sham density functional theory. And the PAW atomic dataset plays an essential role in the implementation and application of the method. The intensive use of proprietary datasets with limited metadata in previous years has led to difficulties in both the cross-validation of PAW codes and the understanding of the accuracy and transferability of PAW atomic data. In this work, we focus on the open-source ABINIT Jollet–Torrent–Holzwarth (JTH) dataset library in the PAW-XML format and investigate the implementation techniques to clarify how the atomic data participate in the computations. We propose an intermediate dataset that extends the original PAW-XML one by atomic quantities in derived forms, which facilitate the PAW implementation using the JTH library and cover the differences between PAW datasets. Our implementation is validated by comparing the structural property results of representative bulk materials and molecules with those calculated by ABINIT using the same datasets. Moreover, we discuss the feasibility of using the intermediate dataset for a quick support of PAW-XML datasets in existing PAW code.

The goal of this paper is to propose fully discrete local discontinuous Galerkin (LDG) finite element methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equation, which are shown to be unconditionally energy stable. In details, using the convex splitting principle, we first construct a first order scheme and a second order Crank-Nicolson scheme for time discretizations. The proposed schemes are shown to be unconditionally energy stable. Then, using the invariant energy quadratization (IEQ) approach, we develop a novel linear and decoupled first order scheme, which is easy to implement and energy stable. In addition, a semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting scheme is employed to improve the temporal accuracy. Due to the local properties of the LDG methods, the resulting algebraic equations at the implicit level is easy to implement and can be solved in an explicit way when it is coupled with iterative methods. In particular, we present efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the capability and efficiency of the proposed methods.