arXiv subjects

Xianfa Hu

Publications and source records attributed to Xianfa Hu.

6 recordsLinked to original sources

arXiv2026

A posteriori error estimates for the two-step explicit exponential Adams method for parabolic equations

In this paper, we derive optimal order a posteriori error estimates for the variable step-size explicit two-step exponential Adams (E-Adams2) method for parabolic problems. We begin by introducing an E-Adams2 approximation, deffned by the piecewise linear approximate solutions, which leads to suboptimal error estimates. To recover optimal order error estimates, we introducean appropriate reconstruction of the approximation with the second order residual for the explicit E-Adams2 method, which plays key roles in deriving optimal order a posteriori error estimates for the proposed explicit method for linear and semilinear parabolic equations. Various numerical experiments are carried out to verify the correct convergence rates of the a posteriori quantities, and the high efffciency of the adaptive algorithm.

math.NA↗

arXiv2026

An adaptive integrating factor midpoint method for second order evolution equations

In this paper, we consider the integrating factor midpoint method for wave-type equations and derive optimal order a posteriori error estimates. We first introduce an integrating factor midpoint approximation defined by the piecewise linear approximate solutions, and derive suboptimal order residual-based error estimates using the energy technique. Hence the key is introducing a continuous, piecewise quadratic time reconstruction to establish optimal order error bounds. Based on the reliable a posteriori error control, we develop an adaptive time-stepping strategy. Numerical examples are implemented to verify the convergence rate of an error estimator and the high efficiency of the adaptive algorithm.

math.NA↗

arXiv2024

A posteriori error estimates for the exponential midpoint method for linear and semilinear parabolic equations

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.

math.NA↗

arXiv2023

Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems

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.

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arXiv2022

Two new families of fourth-order explicit exponential Runge--Kutta methods with four stages for first-order differential systems

In this paper, two new families of fourth-order explicit exponential Runge--Kutta (ERK) methods with four stages are studied for solving first-order differential systems $y'(t)+My(t)=f(y(t))$. By comparing the Taylor series of the exact solution, the order conditions of these ERK methods are derived, which are exactly identical to the order conditions of explicit Runge--Kutta methods, and these ERK methods reduce to classical Runge--Kutta methods once $M\rightarrow \mathbf{0}$. Moreover, we analyze the stability properties and the convergence of the new methods. Several numerical examples are implemented to illustrate the accuracy and efficiency of these ERK methods by comparison with standard exponential integrators.

math.NA↗

arXiv2022

Two new classes of exponential Runge-Kutta integrators for efficiently solving stiff systems or highly oscillatory problems

We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes of exponential Runge-Kutta (ERK) integrators for efficiently solving stiff systems or highly oscillatory problems. We first present a novel class of explicit modified version of exponential Runge-Kutta (MVERK) methods based on the order conditions. Furthermore, we consider a class of explicit simplified version of exponential Runge-Kutta (SVERK) methods. Numerical results demonstrate the high efficiency of the explicit MVERK integrators and SVERK methods derived in this paper compared with the well-known explicit ERK integrators for stiff systems or highly oscillatory problems in the literature.

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