Applied Numerical Mathematics最新文献

Numerical simulations of convective solid-liquid phase change problems have long been a complex problem due to the movement of the solid-liquid interface layer, which leads to a free boundary problem. This work develops a convective phase change heat transfer model based on the phase field method. The governing equations consist of the incompressible Navier-Stokes-Boussinesq equations, the heat transfer equation, and the Allen-Cahn equation. The Navier-Stokes equations are penalised for imposing zero velocity within the solid region. For numerical methods, the mini finite element approach (P1b-P1) is used to solve the momentum equation spatially, the temperature and the phase field are approximated by the P1b elements. In the temporal discretization, the phase field and the temperature are decoupled from the momentum equation by using the finite difference method, forming a solvable linear system. A maximum bound principle for the phase field is derived, coming with an estimation of the tolerance of the time step size, which depends on the temperature range. This estimation guides the time step choice in the simulation. The program is developed within the FreeFem++ framework, drawing on our previous work on phase field methods [1] and a mushy-region method toolbox for heat transfer [2]. The accuracy and effectiveness of the proposed method have been validated through real-world cases of melting and solidification with linear or nonlinear buyangcy force, respectively. The simulation results are in agreement with experiments in references.

In the present study, we construct a considerably fast convergent multistep collocation technique in order to solve Volterra integral equations, especially first-kind ones with variable vanishing delays. Through a robust theoretical analysis, the optimal global convergence of the numerically achieved solutions to their exact counterparts has been demonstrated with the corresponding high orders. The allusion to the strategy of reformulating a first-kind Volterra integral equation into a second-kind Volterra functional integral equation, assists us for the establishment of regularity, existence and uniqueness features of analytical solution over under consideration equation. The existence and uniqueness of numerical solution have also been shown. Eventually, some test problems have been provided to evaluate effectiveness of the proposed multistep collocation technique.

This paper proposes a new pair of block techniques (NPBT) for the direct solution of third-order singular initial-value problems (IVPs). The proposed method uses a polynomial and two intermediate points to approximate the theoretical solution of third-order singular IVPs, resulting in a reasonable approximation within the integration interval. The method's essential features, including stability and convergence order, are analyzed. The proposed NPBT method is improved by using an embedding-like strategy that allows it to be executed in a variable step size mode in order to gain better efficiency. The effectiveness of the proposed method is assessed using various model problems. The approximate solution provided by the proposed NPBT method is more accurate than that of the existing methods utilized for comparison. This efficient solution positions NPBT as a good numerical method for integrating third-order singular IVP models in the fields of applied sciences and engineering.

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests.

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval $[0,b]$. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on $(0,b]$, with possible singularities of the derivatives of the solution at the left endpoint of the interval $[0,b]$.

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order $O\left(N^{-(k-1/2)}\right)$ on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order $O\left({(N^{-1}\ln N)}^{(k-1/2)}\right)$ on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

In this paper, we develop two discontinuous Galerkin (DG) finite element methods to solve the linear poroelasticity in the total pressure formulation, where displacement, fluid pressure, and total pressure are unknowns. The fully-discrete standard DG and conforming DG methods are presented based on the discontinuous approximations in space and the implicit Euler discretization in time. Compared to the standard DG method with penalty terms, the conforming DG method removes all stabilizers and maintains conforming finite element formulation by utilizing weak operators defined over discontinuous functions. The two methods provide locally conservative solutions and achieve locking-free properties in poroelasticity. We also derive the well-posedness and optimal a priori error estimates, which show that our methods satisfy parameter-robustness with respect to the infinitely large Lamé constant and the null-constrained specific storage coefficient. Several numerical experiments are performed to verify these theoretical results, even in heterogeneous porous media.

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form $C^{n+1}=C^n+\Delta t\varphi \left(x,t,C^n,F\left(u^n\right)\right)$. Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.