Applied Numerical Mathematics最新文献

In this work, we present approaches to rigorously certify A- and $A\left(\alpha \right)$-stability in Runge-Kutta methods through the solution of convex feasibility problems defined by linear matrix inequalities. We adopt two approaches. The first is based on sum-of-squares programming applied to the Runge-Kutta E-polynomial and is applicable to both A- and $A\left(\alpha \right)$-stability. In the second, we sharpen the algebraic conditions for A-stability of Cooper, Scherer, Türke, and Wendler to incorporate the Runge-Kutta order conditions. We demonstrate how the theoretical improvement enables the practical use of these conditions for certification of A-stability within a computational framework. We then use both approaches to obtain rigorous certificates of stability for several diagonally implicit schemes devised in the literature.

A BDM type of $H\left(\mathrm{div}\right)$ mixed finite element is constructed on polygonal and polyhedral meshes. The flux space is the $H\left(\mathrm{div}\right)$ subspace of the n-product ${\Pi }_i{P}_k{\left(T_i\right)}^d$ space such that the divergence is a one-piece $P_{k-1}$ polynomial on the big polygon or polyhedron T. Here we assume the 2D polygon can be subdivided into triangles by connecting only one vertex with some vertices of the polygon. For the 3D polyhedron we assume it can be subdivided into tetrahedra, with no added vertex on subdividing its face-polygons, and with either no internal edge or one internal edge. Such mixed finite elements can be more economic on quadrilateral and hexahedral meshes, compared with the standard BDM mixed element on triangular and tetrahedral meshes. Numerical tests and comparisons with the triangular and tetrahedral BDM finite elements are provided.

In this paper we propose an approximation method based on the classical Schoenberg-Marsden variation diminishing operator with applications to the construction of new quadrature rules. We compare the new operator with the multilevel one studied in [12] in order to characterize both of them with respect to the well known classical one. We discuss convergence properties and present numerical experiments.

The Neumann–Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann–Neumann methods that have better convergence properties and require fewer computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov–Poincaré operators.

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.