• Abstract

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time.

• General Summary

• Abstract

Using newly developed $\mathbf{H}$(${\rm curl}^2$) conforming elements, we solve the Maxwell’s transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the $\mathbf{H}$(${\rm curl}^2$))-norm and $\mathbf{H}$(${\rm curl}$)-semi-norm) are established. Numerical experiments are performed to verify the theoretical assumptions and confirm our theoretical analysis.

• General Summary

• Abstract

In this paper, we present the Gauss’s law-preserving spectral methods and their efficient solution algorithms for curl-curl source and eigenvalue problems arising from Maxwell’s equations. Arbitrary order $\boldsymbol{H}(curl)$-conforming spectral basis functions in two and three dimensions are firstly proposed using compact combination of Legendre polynomials. A mixed formulation involving a Lagrange multiplier is then adopted to preserve the Gauss’s law in the weak sense. To overcome the bottleneck of computational efficiency caused by the saddle-point nature of the mixed scheme, we present highly efficient algorithms based on reordering and decoupling of the linear system and numerical eigen-decomposition of 1D mass matrix. The proposed solution algorithms are direct methods requiring only several matrix-matrix or matrix-tensor products of $N$-by-$N$ matrices, where $N$ is the highest polynomial order in each direction. Compared with other direct methods, the computational complexities are reduced from $\mathcal{O}(N^6)$ and $\mathcal{O}(N^9)$ to $\mathcal{O}(N^{log_2 7})$ and $O(N^{1+log^2 7})$ with small and constant pre-factors for 2D and 3D cases, respectively. Moreover, these algorithms strictly obey the Helmholtz-Hodge decomposition, thus totally eliminate the spurious eigen-modes of non-physical zero eigenvalues for convex domains. Ample numerical examples for solving Maxwell’s source and eigenvalue problems are presented to demonstrate the accuracy and efficiency of the proposed methods.

• General Summary

• Abstract

A two-layer dual strategy is proposed in this work to construct a new family of high-order finite volume element (FVE-2L) schemes that can avoid main common drawbacks of the existing high-order finite volume element (FVE) schemes. The existing high-order FVE schemes are complicated to construct since the number of the dual elements in each primary element used in their construction increases with a rate $\mathcal{O}((k+1)^2 ),$ where k is the order of the scheme. Moreover, all k-th-order FVE schemes require a higher regularity $H^{k+2}$ than the approximation theory for the $L^2$ theory. Furthermore, all FVE schemes lose local conservation properties over boundary dual elements when dealing with Dirichlet boundary conditions. The proposed FVE-2L schemes has a much simpler construction since they have a fixed number (four) of dual elements in each primary element. They also reduce the regularity requirement for the $L^2$ theory to $H^{k+1}$ and preserve the local conservation law on all dual elements of the second dual layer for both flux and equation forms. Their stability and $H^1$ and $L^2$ convergence are proved. Numerical results are presented to illustrate the convergence and conservation properties of the FVE-2L schemes. Moreover, the condition number of the stiffness matrix of the FVE-2L schemes for the Laplacian operator is shown to have the same growth rate as those for the existing FVE and finite element schemes.

• General Summary

• Abstract

The linear stabilization approach is well-known for facilitating the use of large time steps in solving gradient flows while maintaining stability. However, the up-to-date analysis of energy stability relies on either a global Lipschitz nonlinearity or an $ℓ^∞$ bound assumption of numerical solutions. Considering the Swift-Hohenberg equation that lacks a global Lipschitz nonlinearity, we develop a unified framework to analyze the energy stability and characterize the stabilization size for a class of single-step schemes employing spatial Fourier pseudo-spectral discretization. First, assuming that all stage solutions are bounded in the $ℓ^∞$ norm, we illustrate that the energy obtained from a single-step scheme with non-negative energy-stability-preserving coefficient is unconditionally dissipative, as long as a sufficiently large stabilization parameter is employed. To justify the $ℓ^∞$ bound assumption of solutions, we use the third-order exponential-time-differencing Runge-Kutta scheme as a case study to establish a uniform-in-time discrete $H^2$ bound for stage solutions under an $\mathcal{O}(1)$ time step constraint. This leads to a uniform $ℓ^∞$ bound of stage solutions through discrete Sobolev embedding. Consequently, we achieve a stabilization parameter of $\mathcal{O}(1),$ which is independent of the time step, thereby ensuring the energy stability. The global-in-time energy stability analysis and characterization of the stabilization parameter represent significant advancements for general single-step schemes applied to a gradient flow without the global Lipschitz continuity.

• General Summary

• Abstract

The time-space nonlocal evolution equations are powerful implementation for modeling anomalous diffusion. In this research, we study the nonlocal nonautonomous reaction-diffusion equation

where $\chi$ is a Lusin space, $∂^w_t$ is a generalized time fractional derivative, $κ$ is a bounded reaction rate, and $\mathcal{L}$ is an infinitesimal generator in terms of semigroup induced by a symmetric Markov process $X.$ We show that the stochastic representation $u(t,x)$ defined by $$u(t,x)=\mathbb{E}^x[e^{\int^t_0\kappa(r,X_{E_t-E_r})dE_r}f(X_{E_t})]$$ is the unique mild as well as weak solution. By further analysis, one can get that the above stochastic representation is also the unique strong solution, and the higher spatial and temporal regularity are obtained. In some particular cases, the corresponding dynamical behaviors are displayed.

• General Summary

• Abstract

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e. on arbitrary meshes. Numerical tests confirm the theory.

• General Summary

• Abstract

Saddle points largely exist in complex systems and play important roles in various scientific problems. High-index saddle dynamics (HiSD) is an efficient method for computing any-index saddle points and constructing solution landscape. In this paper, we propose a two-step Adams explicit scheme for HiSD and analyze its error estimate versus time step. Through careful argumentation and overcoming the difficulties caused by nonlinear coupling and orthogonalisation, we prove that the two-step Adams explicit scheme has second-order accuracy. The theoretical results are further verified by two numerical experiments.

• General Summary

• Abstract

Variable-exponent fractional models attract increasing attentions in various applications, while rigorous mathematical and numerical analysis for typical models remains largely untreated. This work provides general tools to address these models. Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the subdiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractional boundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove their well-posedness. The relations between the convolution method and the perturbation method are discussed, and we further apply the latter to prove the well-posedness of the variable-exponent Abel integral equation and discuss the constraint on the data under different initial values of variable exponent.

• General Summary

• Abstract

We establish a sharp uniform-in-time error estimate for the stochastic gradient Langevin dynamics (SGLD), which is a widely-used sampling algorithm. Under mild assumptions, we obtain a uniform-in-time $\mathcal{O}(η^2)$ bound for the Kullback-Leibler divergence between the SGLD iteration and the Langevin diffusion, where $η$ is the step size (or learning rate). Our analysis is also valid for varying step sizes. Consequently, we are able to derive an $\mathcal{O}(η)$ bound for the distance between the invariant measures of the SGLD iteration and the Langevin diffusion, in terms of Wasserstein or total variation distances. Our result can be viewed as a significant improvement compared with existing analysis for SGLD in related literature.

• General Summary

• Abstract

In this paper, we propose an efficient iterative method called RB-iteration, based on reduced basis (RB) techniques, for addressing time-dependent problems with random input parameters. This method reformulates the original model such that the left-hand side is parameter-independent, while the right-hand side remains parameter-dependent, facilitating the application of fixed-point iteration for solving the system. High-fidelity simulations for time-dependent problems often demand considerable computational resources, rendering them impractical for many applications. RB-iteration enhances computational efficiency by executing iterations in a reduced order space. This approach results in significant reductions in computational costs. We conduct a rigorous convergence analysis and present detailed numerical experiments for the RB-iteration method. Our results clearly demonstrate that RB-iteration achieves superior efficiency compared to the direct fixed-point iteration method and provides enhanced accuracy relative to the classical proper orthogonal decomposition (POD) greedy method.

• General Summary

• Abstract

In many regions of the world, languages coexist in daily life, but often one tongue increases its use at the expense of another. In the present paper, we build a large compartmental system of differential equations that meets the situation of two “prestigious” tongues and many local languages, whose use is reduced by social interaction. The focus is on the preferred language in social relationships for communicating, rather than mere knowledge. We aim at stating and proving theorems on the qualitative behavior of the system. Numerical simulations illustrate the results, giving rise to distinct dynamics.

• General Summary