arXiv subjects

Xue-song Li

Publications and source records attributed to Xue-song Li.

11 recordsLinked to original sources

arXiv2025

Asymptotic behavior for a finitely degenerate semilinear pseudo-parabolic equation

This paper investigates the initial boundary value problem of a finitely degenerate semilinear pseudo-parabolic equation associated with Hörmander's operator. Based on the global existence of solutions in previous literature, the exponential decay estimate of the energy functional is obtained. Moreover, by developing some novel estimates about solutions and using the energy method, the upper bounds of both blow-up time and blow-up rate and the exponential growth estimate of blow-up solutions are determined. In addition, the lower bound of blow-up rate is estimated when a finite time blow-up occurs. Finally, it is established that as time approaches infinity, the global solutions strongly converge to the solution of the corresponding stationary problem. These results complement and improve the ones obtained in the previous literature.

math-ph↗

arXiv2025

Qualitative properties of solutions to a fractional pseudo-parabolic equation with singular potential

This paper investigates the initial boundary value problem for a fractional pseudo-parabolic equation with singular potential. The global existence and blow-up of solutions to the initial boundary value problem are obtained at low initial energy. Moreover, the exponential decay estimates for global solutions and energy functional are further derived, and the upper and lower bounds of both blow-up time and blow-up rate for blow-up solutions are respectively estimated. Specifically, we extend the method for proving blow-up of solutions with negative initial energy in previous literatures to cases involving nonnegative initial energy, which broadens the applicability of this method. Finally, for the corresponding stationary problem, the existence of ground-state solutions is established, and it is proved that the global solutions strongly converge to the solutions of stationary problem as time tends to infinity.

math.OC↗

arXiv2024

Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem

This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving the unconstrained optimization problem. Under certain conditions, we demonstrate that this continuous-time dynamical system exponentially converges to the optimal solution of the inequality constrained convex optimization problem. Moreover, we derive several discrete-time algorithms from this continuous-time framework and obtain their optimal convergence rates. Finally, we present numerical experiments to validate the effectiveness of the proposed algorithms.

math.OC↗

arXiv2023

Finite-time and fixed-time consensus control of multi-agent systems driven by parabolic partial differential equations

This paper focuses on the study of the finite-time consensus (FTC) and fixed-time consensus (FXC) issues of multi-agent systems (MASs) driven by parabolic partial differential equations (PDEs). Compared with the study in the existing literature, the topic of FTC and FXC control is first embodied in MASs driven by parabolic PDEs. Based on the Lyapunov theorems, the FTC and FXC controllers are devised to ensure that the MASs converge to a stable state with external disturbance. Furthermore, we simplify the controllers to guarantee the FTC and FXC of MASs without external disturbance. Finally, two illustrative examples are given to verify the feasibility of controllers.

math.OC↗

arXiv2022

Stochastic differential variational inequalities with applications

In this paper, we introduce and study a stochastic differential variational inequality (SDVI) which consists of a stochastic differential equation and a stochastic variational inequality. We obtain the existence and uniqueness of the solutions for SDVI by using the iteration method and Gronwall's inequality. Moreover, we show the convergence of Euler scheme for solving SDVI under some mild conditions. Finally, we apply the obtained results to solve the electrical circuits with diodes and the collapse of the bridge problems in stochastic environment.

math.OC↗

arXiv2018

An Improved Roe Scheme for All Mach-Number Flows Simultaneously Curing Known Problems

Roe scheme is known for its good performance in moderate-Mach-number flows. However, this scheme and its extended versions suffers from many disastrous problems, such as non-physical behavior, global cut-off, and checkerboard problems, for incompressible flows; and shock instability, expansion shock, and positively non-conservative problems for hypersonic flows. In this paper, non-physical behavior problem, checkerboard problem, and main reason of shock instability problem are due to that the Roe scheme cannot identify multi-dimensional incompressible and compressible flows when normal Mach number on the cell face tends to zero, and then leads to incorrect cross modifications. Positively non-conservative problem is also identified as another important reason for shock instability. Therefore, Mach number and an assistant pressure-density-varying detector are introduced into the Roe scheme to judge compressibility, positivity condition is satisfied by a simple modification with minimal numerical dissipation increases and even with possible decreases in numerical dissipation, the mechanism of the preconditioned Roe scheme is introduced to suppress checkerboard problem, and modified entropy fix and the rotated Riemann solver is combined with complementary advantages as an assistant improvement for better robust. Based on above improvements and previous developments for global cut-off and expansion shock problems, an improvement Roe scheme for all Mach-number flow (Roe-AM) is proposed to simultaneously overcome nearly all well-known drawbacks of the classical Roe scheme. The Roe-AM scheme is simple, easy to implement, computationally low-cost, robust, good extensibility, and free of empirical parameters essentially, with increasing minimal numerical dissipation.

physics.comp-ph↗

arXiv2016

The Role of Momentum Interpolation Mechanism of the Roe Scheme in the Shock Instability

The shock instability phenomenon is a famous problem for the shock-capturing scheme. By subdividing the numerical dissipation of the Roe scheme, the term of pressure-difference-driven modification for the cell face velocity is regarded as a version of the momentum interpolation method (MIM), which is necessary for incompressible flows to suppress the pressure checkerboard problem. Through the analysis and odd-even decoupling test, it is discovered that MIM plays the most important role on the shock instability. In fact, for non-linear flows MIM should be completely removed, but unexpected MIM is activated on the cell face nearly parallel to the flow for high Mach number flows or low Mach number flows in shock. For such conditions, two coefficients are designed based on local Mach number and a shock detector, respectively, and then the improved Roe scheme is proposed, which gives consideration to requirement of MIM for incompressible and compressible flows and is validated for good performance of avoiding odd-even decoupling. Therefore, the aim of decreasing rather than increasing numerical dissipation to cure the shock instability can be achieved.

physics.flu-dyn↗

arXiv2016

Cures for the Expansion Shock and the Shock Instability of the Roe Scheme

A common defect of the Roe scheme is the production of non-physical expansion shock and shock instability. An improved method with several advantages was presented to suppress the shock instability. However, this method cannot prevent expansion shock and is incompatible with the traditional curing method for expansion shock. Therefore, the traditional curing mechanism is analyzed. The discussion explains the effectiveness of the traditional curing method and identifies several defects, one of which leads to incompatibility between curing the shock instability and expansion shock. Consequently, a new improved Roe scheme is proposed in this study. This scheme is concise, easy to implement, low computational cost, and robust. More importantly, the scheme can simultaneously cure the shock instability and expansion shock without additional costs.

physics.flu-dyn↗

arXiv2015

On the All-Speed Roe-type Scheme for Large Eddy Simulation of Homogeneous Decaying Turbulence

As the representative of the shock-capturing scheme, the Roe scheme fails to LES because important turbulent characteristics cannot be reproduced such as the famous k-5/3 spectral law owing to large numerical dissipation. In this paper, the Roe scheme is divided into five parts: , , , , and , which means basic upwind dissipation, pressure-difference-driven and velocity-difference-driven modification of the interface fluxes and pressure, respectively. Then, the role of each part on LES is investigated by homogeneous decaying turbulence. The results show that the parts , , and have little effect on LES. It is important especially for because it is necessary for computation stability. The large numerical dissipation is due to and , and each of them has much larger dissipation than SGS dissipation. According to these understanding, an improved all-speed LES-Roe scheme is proposed, which can give enough good LES results for even coarse grid resolution with usually adopted reconstruction.

physics.flu-dyn↗

arXiv2013

A Uniform Algorithm for All-Speed Shock-Capturing Schemes

There are many ideas for developing shock-capturing schemes and their extension for all-speed flow. The representatives of them are Roe, HLL and AUSM families. In this paper, a uniform algorithm is proposed, which expresses three families in the same framework. The algorithm has explicit physical meaning, provides a new angel of understanding and comparing the mechanism of schemes, and may play a great role in the further research. As an example of applying the uniform algorithm, the low-Mach number behaviour of the schemes is analyzed. Then, a very clear and simple explanation is given based on the wall boundary, and a concise rule is proposed to judge whether a scheme has satisfied low-Mach number behaviour.

physics.comp-ph↗

arXiv2011

On the Mechanism of Roe-type Schemes for All-Speed Flows

In recent years, Roe-type schemes based on different ideas have been developed for all-speed flows, such as the preconditioned Roe, the All-Speed Roe, Thornber's modified Roe and the LM-Roe schemes. This work explores why these schemes succeed or fail with the accuracy and checkerboard problems. Comparison and analysis show that the accuracy and checkerboard problems are caused by the order of the sound speed being too large and too small in the coefficients of the velocity-derivative and pressure-derivative dissipation terms, respectively. These problems can be resolved by choosing coefficients with zero-order sound speed. In addition, to avoid the negative effects of the global cut-off strategy on accuracy while maintaining computational stability, the sound speed terms in the numerator of the coefficients can be determined by local variables, while those in the denominator remain the global cut-off. Two novel schemes are proposed as examples to demonstrate how these ideas can be applied to construct more satisfactory schemes for all-speed flows. Asymptotic analysis and numerical experiments support the theoretical analysis and the rules obtained in the work.

math.NA↗