Search

In this paper, the a posteriori error estimates of the exponential midpoint method for time discretization are studied for linear and semilinear parabolic equations. Using the exponential midpoint approximation defined by a continuous and piecewise linear interpolation of nodal values yields the suboptimal order estimates. Based on the property of the entire function, we introduce a continuous and piecewise quadratic time reconstruction of the exponential midpoint method to derive the optimal order estimates, and the error bounds are solely dependent on the discretization parameters, the data of the problem and the approximation of the entire function. Several numerical examples are implemented to illustrate the theoretical results.

The Landau Fermi liquid theory, a cornerstone in condensed matter physics, encounters limitations in explaining certain phenomena, like the peculiar behavior of strange metals in high-temperature superconductors. Non-Fermi liquids, like Bose metals with uncondensed bosonic ground state, offer potential explanations, yet constructing an elusive Bose metal phase in two dimensions (2D) remains a formidable challenge. Utilizing constraint path quantum Monte Carlo and functional renormalization group methods on a fermionic system with spin anisotropy in a 2D lattice, we reveal the emergence of a Cooper pair Bose metal in a highly anisotropic regime (a < 0.30) with wide range of filling, most notably at a filling fraction of n~0.8. Our findings exhibit a visible nonzero momentum Bose surface in the Cooper-pair distribution function, accompanied by a distinct signal of dxy correlation between pairs. Our results highlight that spin-dependent anisotropy in the Fermi surface leads to versatile pairing forms. Platforms such as ultracold atoms in optical lattices and recently proposed altermagnets hold promise for realizing this intriguing phase., Comment: 15 pages, 13 figures

We derive optimal order a posteriori error estimates in the $L^\infty(L^2)$ and $L^1(L^2)$-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the $L1$ methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results., Comment: 22 pages

In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a first-order symplectic method. In order to solve highly oscillatory problems, the highly accurate implicit ERK integrators (up to order four) are formulated by comparing the Taylor expansions of numerical and exact solutions, it is shown that the order conditions of two new kinds of exponential methods are identical to the order conditions of classical Runge-Kutta (RK) methods. Moreover, we investigate the linear stability properties of these exponential methods. Finally, numerical results not only present the long time energy preservation of the first-order symplectic method, but also illustrate the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.

We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. In this novel algorithm, a pair of deep neural networks for the approximations of the gradient and the integral kernel is introduced in a crucial way based on deep FBSDE method. To derive the error estimates for this deep learning algorithm, the convergence of Markovian iteration, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated. Two numerical examples are provided to show the efficiency of this proposed algorithm.

In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in $H^{-1}$ and $\mathcal H^s_\alpha$ ($s=1,2,3$) spaces independent of the initial data and time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes.The uniform $\mathcal H^3_\alpha$ bound is obtained on basis of the uniform $H^1$ estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation., Comment: 25 pages, 2 figures

In this paper stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the $l^\infty(0,T;H)$-stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The $l^\infty(0,T;V)$-stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.

In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size $H$ and the fine grid size $h$ satisfy $H=O(h^{\frac{r-1}{r}})$. Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.