In this paper, we investigate the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear Korteweg-de Vries type equations. The numerical flux for the nonlinear convection term is chosen as the generalized Lax-Friedrichs flux, and the generalized alternating flux and upwind-biased flux are used for the dispersion term. The generalized Lax-Friedrichs flux with anti-dissipation property will compensate the numerical dissipation of the dispersion term, resulting in a nearly energy conservative scheme that is useful in resolving waves and is beneficial for long time simulations. To deal with the nonlinearity and different numerical flux weights, a suitable numerical initial condition is constructed, for which a modified global projection is designed. By establishing relationships between the prime variable and auxiliary variables in combination with sharp bounds for jump terms, optimal error estimates are obtained. Numerical experiments are shown to confirm the validity of theoretical results.

In this paper, we consider numerical solutions of the fractional diffusion equation with the order time fractional derivative defined in the Caputo-Hadamard sense. A high order time-stepping scheme is constructed, analyzed, and numerically validated. The contribution of the paper is twofold: 1) regularity of the solution to the underlying equation is investigated, 2) a rigorous stability and convergence analysis for the proposed scheme is performed, which shows that the proposed scheme is 3 + order accurate. Several numerical examples are provided to verify the theoretical statement.

In practical applications, the Allen-Cahn (AC) equation is commonly used to model microstructure evolutions, including alloy solidification, crystal growth, fingerprint image restoration, and image segmentation. However, when we discretize the AC equation with a conventional finite difference scheme, the directional bias in error terms introduces anisotropy into the numerical results, affecting interface dynamics. To address this issue, we use two- and three-dimensional isotropic finite difference schemes to solve the AC equation. Stability of the proposed algorithm is verified by deriving the time step constraints in both 2D and 3D domains. To demonstrate the sharp estimation of the stability constraints, we conducted several numerical experiments and found the maximum principle is guaranteed under the analyzed time-step constraint.

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous cavity. We shall develop a modified factorization method to reconstruct the shape and location of the interior interface of the inhomogeneous cavity by means of many internal measurements of the near-field data. Numerical examples are carried out to illustrate the practicability of the inversion algorithm.

Stochastic gradient descent (SGD) methods have gained widespread popularity for solving large-scale optimization problems. However, the inherent variance in SGD often leads to slow convergence rates. We introduce a family of unbiased stochastic gradient estimators that encompasses existing estimators from the literature and identify a gradient estimator that not only maintains unbiasedness but also achieves minimal variance. Compared with the existing estimator used in SGD algorithms, the proposed estimator demonstrates a significant reduction in variance. By utilizing this stochastic gradient estimator to approximate the full gradient, we propose two mini-batch stochastic conjugate gradient algorithms with minimal variance. Under the assumptions of strong convexity and smoothness on the objective function, we prove that the two algorithms achieve linear convergence rates. Numerical experiments validate the effectiveness of the proposed gradient estimator in reducing variance and demonstrate that the two stochastic conjugate gradient algorithms exhibit accelerated convergence rates and enhanced stability.

We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck (PNP) equations, which are a nonlinear coupled system widely used in semiconductors and ion channels. After presenting the semi-discrete scheme, the optimal $H^1$ norm error estimates are presented for the time-dependent PNP equations, which are based on some error estimates of a virtual element energy projection. The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation. The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.

This paper presents space-time continuous and time discontinuous Galerkin schemes for solving nonlinear time-fractional partial differential equations based on B-splines in time and non-uniform rational B-splines (NURBS) in space within the framework of Isogeometric Analysis. The first approach uses the space-time continuous Petrov-Galerkin technique for a class of nonlinear time-fractional Sobolev-type equations and the optimal error estimates are obtained through a concise equivalence analysis. The second approach employs a generalizable time discontinuous Galerkin scheme for the time-fractional Allen-Cahn equation. It first transforms the equation into a time integral equation and then uses the discontinuous Galerkin method in time and the NURBS discretization in space. The optimal error estimates are provided for the approach. The convergence analysis under time graded meshes is also carried out, taking into account the initial singularity of the solution for two models. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methods.

A fractional-order mathematical model of lung cancer is used to describe the dynamics of tumor growth and the interactions between cancer cells and immune cells. To obtain approximate solutions and better understand the behavior of the state functions, a pseudo-operational collocation scheme employing shifted Jacobi polynomials as basis functions is introduced. Initially, the existence and uniqueness of solutions to the model are established using the Leray-Schauder fixed-point theorem. Error bounds for the residual functions are estimated within a Jacobi-weighted $L^2$-space. To enhance the accuracy and reliability of the results, two distinct strategies are implemented: sensitivity analysis and feedback control. The feedback control of the proposed pseudo-operational spectral method is performed using the method of Lagrange multipliers, marking its first application in this context. Spectral solutions are derived by applying the pseudo-operational scheme to both the original model and the model with control functions. Improved performance and outputs are anticipated following the application of the feedback control strategy. Finally, comprehensive biological interpretations of the results are provided, offering insights into the practical implications of the model.

This paper presents three regularized models for the logarithmic Klein-Gordon equation. By using a modified Crank-Nicolson method in time and the Galerkin finite element method (FEM) in space, a fully implicit energy-conservative numerical scheme is constructed for the local energy regularized model that is regarded as the best one among the three regularized models. Then, the cut-off function technique and the time-space error splitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any coupling condition between the temporal step size and the spatial mesh width. The theoretical framework is uniform for the other two regularized models. Finally, numerical experiments are provided to verify our theoretical results. The analytical techniques in this work are not limited in the FEM, and can be directly extended into other numerical methods. More importantly, this work closes the gap for the unconditional error/stability analysis of the numerical methods for the logarithmic systems in higher dimensional spaces.

Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions’ qualitative properties seems limited, partially due to the lack of efficient and reliable numerical methods. In this paper, we design a dedicated numerical method to explore these nontrivial solutions further. We first design an improved adaptive orthogonal basis deflation method by combining the adaptive orthogonal basis method with a bisection-deflation algorithm. We then apply the proposed new method to study the impact of domain changes on multiple solutions of certain nonlinear elliptic equations. When the domain varies from a circular disk to an elliptical disk, the corresponding functional value changes dramatically for some particular solutions, which indicates that these nontrivial solutions in the circular domain may become unstable in the elliptical domain. Moreover, several theoretical results on multiple solutions in the existing literature are verified. For the nonlinear sine-Gordon equation with parameter $λ,$ nontrivial solutions are found for $λ > λ_2,$ here $λ_2$ is the second eigenvalue of the corresponding linear eigenvalue problem. For the singularly perturbed Ginzburg-Landau equation, highly concentrated solutions are numerically verified, suggesting that their convergent limit is a delta function when the perturbation parameter goes to zero.