Applied Numerical Mathematics最新文献

We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the reliable cell merging algorithm for smooth interfaces to automatically generate the induced mesh for piecewise smooth interfaces. An hp a posteriori error estimate is derived for a new unfitted finite element method whose finite element functions are conforming in each subdomain. Numerical examples illustrate the competitive performance of the method.

In this study, we have derived a thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects. This model accommodates variations in physical properties such as density, viscosity, heat capacity, and thermal conductivity between the two components. The model equations encompass a Cahn-Hilliard equation with the volume fraction as the phase variable, a Navier-Stokes equation, and a heat equation, and meanwhile maintains mass conservation, energy conservation, and entropy increase simultaneously. Given the highly coupled and nonlinear nature of the model equations, we developed a semi-decoupled, mass-preserving, and entropy-stable time-discrete numerical method. We conducted several numerical tests to validate both our model and numerical method. Additionally, we have investigated the merging process of two bubbles under non-isothermal conditions and compared the results with those under isothermal conditions. Our findings reveal that temperature gradients influence bubble morphology and lead to earlier merging. Moreover, we have observed that the merging of bubbles slows down with increasing heat Peclect number ${\mathrm{Pe}}_T$ when the initial temperature field increases linearly along the channel, while bubbles merge faster with heat Peclect number ${\mathrm{Pe}}_T$ when the initial temperature field decreases linearly along the channel.

In this paper, we consider a class of k-order $(3\le k\le 5)$ backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term ${\tau }_n^k{(-\Delta )}^k{\underset{\_}{D}}_k{\varphi }^n$ (${\underset{\_}{D}}_k$ represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k $(3\le k\le 5)$ methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k $(3\le k\le 5)$ formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the $L^2$ norm stability and convergence of the stabilized BDF-k $(3\le k\le 5)$ schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density $n\in N$ and truncation dimension parameter $M\in N$, is of the order $n^{-1/2}+\delta \left(M\right)$ such that $\delta (\cdot )$ is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported.

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies.

The mathematical modeling of the propagation of diseases has an important role from both mathematical and biological points of view. In this article, we observe an SEIR-type model with a general incidence rate and a non-constant recruitment rate function. First, we observe the qualitative properties of the continuous system and then apply different numerical methods: first-order and higher-order strong stability preserving Runge-Kutta methods. We give different conditions under which the numerical schemes preserve the positivity and the boundedness of the continuous-time solution. Then, the theoretical results are demonstrated by some numerical experiments.

In this paper, we develop a mapped Jacobi spectral Galerkin method for solving the multi-term Fredholm integral equations (MFIEs) with two-sided weakly singularities. We introduce a new family of mapped Jacobi functions (MJFs) and establish the corresponding spectral approximation results on these MJFs in weighted Sobolev spaces involving the mapped Jacobi weight function. These MJFs serve as the basis functions in our algorithm design and are tailored to the two-sided end-points singularities of the solution by using suitable mapping. Moreover, we derive the error estimates of the proposed method for MFIEs. Finally, the numerical examples are provided to demonstrate the accuracy and efficiency of the proposed method.

In this paper, we study a posteriori error estimates for one-dimensional and two-dimensional linear parabolic equations. The backward Euler method and the Crank–Nicolson method for the time discretization are used, and the second-order finite difference method is employed for the space discretization. Based on linear interpolation and interpolation estimate, a posteriori error estimators corresponding to space discretization are derived. For the backward Euler method and the Crank–Nicolson method, the errors due to time discretization are obtained by exploring linear continuous approximation and two different continuous, piecewise quadratic time reconstructions, respectively. As a consequence, the upper and lower bounds of a posteriori error estimates for the fully discrete finite difference methods are derived, and these error bounds depend only on the discretization parameters and the data of the model problems. Numerical experiments are presented to illustrate our theoretical results.

In this paper we propose a high-order spline finite element method for solving a class of time-dependent electromagnetic waves and its associated frequency-domain approach. A Fourier transform and its inverse are used for the time integration of the wave problem. The spatial discretization is performed using a partitioned mesh with tensorial spline functions to form bases of the discrete solution in the variational finite element space. Quadrature methods such as the Gauss-Hermite quadrature are implemented in the inverse Fourier transform to compute numerical solutions of the time-dependent electromagnetic waves. In the present study we carry out a rigorous convergence analysis and establish error estimates for the wave solution in the relevant norms. We also provide a full algorithmic description of the method and assess its performance by solving several test examples of time-dependent electromagnetic waves with known analytical solutions. The method is shown to verify the theoretical estimates and to provide highly accurate and efficient simulations. We also evaluate the computational performance of the proposed method for solving a problem of wave transmission through non-homogeneous materials. The obtained computational results confirm the excellent convergence, high accuracy and applicability of the proposed spline finite element method.

This paper is dedicated to the numerical solution of a mathematical model that describes frictional quasistatic contact between an elastic body and a moving foundation, with the wear effect on the contact interface of the moving foundation due to friction. The mathematical problem is a system consisting of a time-dependent quasi-variational inequality and an integral equation. The numerical method is based on the use of the virtual element method (VEM) for the spatial discretization of the variational inequality and a variable step-size left rectangle integration formula for the integral equation. The existence and uniqueness of a numerical solution are shown, and optimal order error estimates are derived for both the displacement and the wear function for the lowest order VEM. Numerical results are presented to demonstrate the efficiency of the method and to illustrate the numerical convergence orders.