In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.

In this paper, we apply high-order finite difference (FD) schemes for multispecies and multireaction detonations (MMD). In MMD, the density and pressure are positive and the mass fraction of the ith species in the chemical reaction, say zi, is between 0 and 1, with ∑z_i=1. Due to the lack of maximum-principle, most of the previous bound-preserving technique cannot be applied directly. To preserve those bounds, we will use the positivity-preserving technique to all the z′is and enforce ∑z_i=1 by constructing conservative schemes, thanks to conservative time integrations and consistent numerical fluxes in the system. Moreover, detonation is an extreme singular mode of flame propagation in premixed gas, and the model contains a significant stiff source. It is well known that for hyperbolic equations with stiff source, the transition points in the numerical approximations near the shocks may trigger spurious shock speed, leading to wrong shock position. Intuitively, the high-order weighted essentially non-oscillatory (WENO) scheme, which can suppress oscillations near the discontinuities, would be a good choice for spatial discretization. However, with the nonlinear weights, the numerical fluxes are no longer “consistent”, leading to nonconservative numerical schemes and the bound-preserving technique does not work. Numerical experiments demonstrate that, without further numerical techniques such as subcell resolutions, the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence.

Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance.

We develop a new moving-water equilibria preserving numerical scheme for the Saint-Venant system. The new scheme is designed in two major steps. First, the geometric source term is incorporated into the discharge flux, which results in a hyperbolic system with a global flux. Second, the discharge equation is relaxed so that the nonlinearity is moved into the stiff right-hand side of the added auxiliary equation. The main advantages of the new scheme are that (i) no special treatment of the geometric source term is required, and (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables, as it must be done in the existing alternative methods. We also develop a hybrid numerical flux, which helps to handle various flow regimes in a stable manner. Several numerical experiments are performed to verify that the proposed scheme is capable of exactly preserving general moving-water steady states and accurately capturing their small perturbations.

In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson–Nernst–Planck (PNP) equations, which have found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

We present a regularized finite difference method for the logarithmic Schr ̈odingerequation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., ln ρ→ −\infty when ρ→0+v with ρ = |u|^2 being the density and u being the complex-valuedwave function or order parameter, there are significant difficulties in designing numerical methodsand establishing their error bounds for the LogSE. In order to suppress the roundoff error and toavoid blowup, a regularized LogSE (RLogSE) is proposed with a small regularization parameter 0 < ε << 1 and linear convergence is established between the solutions of RLogSE and LogSE interm of ε. Then a semi-implicit finite difference method is presented for discretizing the RLogSEand error estimates are established in terms of the mesh sizehand time stepτas well as the smallregularization parameterε. Finally numerical results are reported to illustrate our error bounds.

We present two uniformly accurate numerical methods for discretizing the Zakharovsystem (ZS) with a dimensionless parameter 0< ε ≤ 1, which is inversely proportional to theacoustic speed. In the subsonic limit regime, i.e., 0< ε << 1, the solution of ZS propagates waves with O(ε)- andO(1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layerswith speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or theincompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, wepresent a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splittingtechnique and an exponential wave integrator for temporal derivatives in the nonlinear Schr ̈odingerequation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, wepropose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicitand convergent exponentially in space for all kinds of initial data, which is uniformly for ε ∈ (0,1].The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accuratein time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accuratein time for any kind of initial data. Extensive numerical results are reported to show their efficiencyand accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied tostudy numerically convergence rates of ZS to its limiting models when ε→0+.

We propose an explicit numerical method for the periodic Korteweg–de Vries equation. Our method is based on a Lawson-type exponential integrator for time integration and the Rusanov scheme for Burgers’ nonlinearity. We prove first-order convergence in both space and time under a mild Courant–Friedrichs–Lewy condition τ=O(h)⁠, where τ and h represent the time step and mesh size for solutions in the Sobolev space H^3((−π,π))⁠, respectively. Numerical examples illustrating our convergence result are given.