We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of four different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.

We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate numerical method for solving a coupled integral equation system. Although the stiffness due to the high order spatial derivatives can be removed using a small scale decomposition technique, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this physically imposed stiffness by employing a rescaling scheme, which accelerates the slow dynamics and reduces the computational cost. Our nonlinear results reveal that positive currents restrain finger ramification and promote the overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no fluid is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law $(t_∗−t)^b,$ where $t_∗$ is the estimated pinch-off time.

We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.

The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics, which is termed entropy condition. However, most cell-centered Lagrangian (CL) schemes do not satisfy the entropy condition. Until 2020, for one-dimensional gas dynamics equations, the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate (AA) flux and the entropy conservative (EC) flux developed by Maire et al. was used. This hybridized CL scheme satisfies the entropy condition; however, it is under-entropic in the part zones of rarefaction waves. Moreover, the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves. Another disadvantage of this scheme is that it is of only first-order accuracy. In this paper, we firstly construct a modified entropy conservative (MEC) flux which can damp effectively numerical oscillations in simulating strong rarefaction waves. Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows. This new hybridized CL scheme is a combination of the AA flux and the MEC flux.
In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic, we propose using the third-order TVD-type Runge-Kutta time discretization method. Based on the new hybridized flux, we develop the second-order CL scheme that satisfies the entropy condition.
Finally, the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.

Thermal phase change problems are widespread in mathematics, nature, and science. They are particularly useful in simulating the phenomena of melting and solidification in materials science. In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase change model, which is the coupling of a heat transfer equation and a phase field equation. The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes. A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step, with an efficient scheme of sufficient accuracy to calculate the solution at the first step. It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps. Adaptive time step size strategies can be applied to further benefit from this unconditional stability. Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

Based on the auxiliary subspace techniques, a posteriori error estimator of nonconforming weak Galerkin finite element method (WGFEM) for Stokes problem in two and three dimensions is presented. Without saturation assumption, we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term. The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix. To reduce the computational cost of the error problem, an equivalent error problem is constructed by using diagonalization techniques, which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom $(d.o.f)$ to get the error estimator. Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

This paper studies the geometric boundary representations for Inverse Lax-Wendroff (ILW) method, aiming to develop a practical computer-aided engineering method without body-fitted meshes. We propose the signed distance function (SDF) representation of the geometric boundary and design an extremely efficient algorithm for foot point calculation, which is particularly in line with the needs of ILW. Theoretical and numerical analyses demonstrate that the SDF representation of geometric boundary can satisfy ILW’s needs better than others. The effectiveness and robustness of our proposed method are verified by simulating initial boundary value computational physical problems of Euler equation for compressible fluids.

The weak Galerkin (WG) method is a nonconforming numerical method for solving partial differential equations. In this paper, we introduce the WG method for elliptic equations with Newton boundary condition in bounded domains. The Newton boundary condition is a nonlinear boundary condition arising from science and engineering applications. We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions. The error estimates are derived. Numerical experiments are presented to verify the theoretical analysis.

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.

We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D. The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function. The Nédélec edge elements are employed to discretize the operators, where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier. The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements. The spectral indicator method is then applied to compute the discrete eigenvalues. Numerical examples are presented demonstrating the effectiveness of the proposed method.