The Cahn–Hilliard equations are a versatile model for describing the evolution of complex morphologies. In this paper we present a computational pipeline for the numerical solution of a ternary phase-field model for describing the nanomorphology of donor–acceptor semiconductor blends used in organic photovoltaic devices. The model consists of two coupled fourth-order partial differential equations that are discretized using a finite element approach. In order to solve the resulting large-scale linear systems efficiently, we propose a preconditioning strategy that is based on efficient approximations of the Schur-complement of a saddle point system. We show that this approach performs robustly with respect to variations in the discretization parameters. Finally, we outline that the computed morphologies can be used for the computation of charge generation, recombination, and transport in organic solar cells.

In this paper, we derive the frozen Gaussian approximation (FGA) for computing the solution to the Dirac equation in curved space in the semi-classical regime. The latter equation is used in particular for modeling electronic scattering on strained graphene surfaces. We present numerical comparisons of the Dirac solutions on curved and flat spaces, illustrating the focusing effect of graphene surfaces, as well as qualitative comparisons with a tight-binding model. A CPU-time comparison shows that FGA becomes more efficient than an IMEX pseudospectral method when the semiclassical parameter is small.

This paper aims at developing a control volume staggered Lagrangian scheme in $r–z$ coordinate that preserves symmetry property. To achieve this goal, the support operator method is first utilized to derive the compatible discretization that satisfies the Geometrical Conservation Law (GCL) and momentum and total energy conservation property. We further introduce a method of source term treatment to recover the spherical symmetry of the current scheme. It is shown that the developed scheme has the benefit of maintaining the momentum and total energy conservation. The equi-angular grid, randomly distorted polar grid, and Cartesian grid are considered for one-dimensional spherical flow simulations. Also, an extension to the non-spherical flow is presented. The results confirm the good performance of the developed scheme.

The heat conduction under fast external excitation exists in many experiments measuring the thermal conductivity in solids, which is described by the phonon Boltzmann equation, i.e., the Callaway’s model with dual relaxation times. Such a kinetic system has two spatial Knudsen numbers related to the resistive and normal scatterings, and one temporal Knudsen number determined by the external oscillation frequency. Thus, it is a challenge to develop an efficient numerical method. Here we first propose the general synthetic iterative scheme (GSIS) to solve the phonon Boltzmann equation, with the fast-converging and asymptotic-preserving properties: (i) the solution can be found within dozens of iterations for a wide range of Knudsen numbers and frequencies, and (ii) the solution is accurate when the spatial cell size in the bulk region is much larger than the phonon mean free path. Then, we investigate how the heating frequency affects the heat conduction in different transport regimes.

In this paper, we construct a two-dimensional third-order space-time conservation element and solution element (CESE) method and apply it to the magnetohydrodynamics (MHD) equations. This third-order CESE method preserves all the favorable attributes of the original second-order CESE method, such as: (i) flux conservation in space and time without using an approximated Riemann solver, (ii) genuine multi-dimensional algorithm without dimensional splitting, (iii) the use of the most compact mesh stencil, involving only the immediate neighboring cells surrounding the cell where the solution at a new time step is sought, and (iv) an explicit, unified space-time integration procedure without using a quadrature integration procedure. In order to verify the accuracy and efficiency of the scheme, several 2D MHD test problems are presented. The result of MHD smooth wave problem shows third-order convergence of the scheme. The results of the other MHD test problems show that the method can enhance the solution quality by comparing with the original second-order CESE scheme.

Based on the idea of serendipity element, we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonal meshes in this article. The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates, and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space. Moreover, we construct a family of unified dual partitions for arbitrary convex polygonal meshes, which is crucial to finite volume element scheme, and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom. Finally, under certain geometric assumption conditions, the optimal $H^1$ error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained, and verified by numerical experiments.

Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifold, using Voronoi tessellation, we propose two upwind finite volume schemes with or without the information of the invariant measure. Both schemes possess stochastic $Q$-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part, enabling us to prove unconditional stability, ergodicity and error estimates. Based on the two upwind schemes, several numerical examples – including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system – are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. This makes them suitable for adapting to manifold-related computations that arise from high-dimensional molecular dynamics simulations.

In rarefied gas flows, the spatial grid size could vary by several orders of magnitude in a single flow configuration (e.g., inside the Knudsen layer it is at the order of mean free path of gas molecules, while in the bulk region it is at a much larger hydrodynamic scale). Therefore, efficient implicit numerical method is urgently needed for time-dependent problems. However, the integro-differential nature of gas kinetic equations poses a grand challenge, as the gain part of the collision operator is non-invertible. Hence an iterative solver is required in each time step, which usually takes a lot of iterations in the (near) continuum flow regime where the Knudsen number is small; worse still, the solution does not asymptotically preserve the fluid dynamic limit when the spatial cell size is not refined enough. Based on the general synthetic iteration scheme for steady-state solution of the Boltzmann equation, we propose two numerical schemes to push the multiscale simulation of unsteady rarefied gas flows to a new boundary, that is, the numerical solution not only converges within dozens of iterations in each time step, but also asymptotically preserves the Navier-Stokes-Fourier limit in the continuum flow regime, when the spatial grid is coarse, and the time step is large (e.g., in simulating the extreme slow decay of two-dimensional Taylor vortex, the time step is even at the order of vortex decay time). The properties of fast convergence and asymptotic preserving of the proposed schemes are not only rigorously proven by the Fourier stability analysis for simplified gas kinetic models, but also demonstrated by several numerical examples for the gas kinetic models and the Boltzmann equation.

This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms. The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally. This work overcomes two difficulties. The first difficulty is the evaluation of singular integrals. Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains, which are solved with FFT-based finite difference solvers. The second one is the expensive computational cost for volume integrals. Despite the use of efficient interface problem solvers, the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem. The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions. Since solving boundary integral equations on a circular boundary is so accurate, one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost. Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.

This paper is devoted to the inverse design of strained graphene surfaces for the control of electrons in the semi-classical optical-like regime. Assuming that charge carriers are described by the Dirac equation in curved-space and exploiting the fact that wave propagation can be described by ray-optics in this regime, a general computational strategy is proposed in order to find strain fields associated with a desired effective refractive index profile. The latter is first determined by solving semi-classical trajectories and by optimizing a chosen objective functional using a genetic algorithm. Then, the graded refractive index corresponding to the strain field is obtained by using its connection to the metric component in isothermal coordinates. These coordinates are evaluated via numerical quasiconformal transformations by solving the Beltrami equation with a finite volume method. The graphene surface deformation is finally optimized, also using a genetic algorithm, to reproduce the desired index of refraction. Some analytical results and numerical experiments are performed to illustrate the methodology.