Wen-Xian Xie

A CWENO-type central-upwind scheme for ideal MHD equations

A simple fourth-order, central-upwind scheme based on central weighted essentially non-oscillatory (CWENO) reconstruction is proposed in this paper for computing the approximate solutions of one- and two-dimensional ideal magnetohydrodynamics (MHD) equations with high-resolution. Since the non-uniform width of the different local Riemann fans is calculated more accurately, the central-upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided. Due to the central-upwind scheme is combined with the fourth-order CWENO reconstruction, the scheme we present has the non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our method.

Path integration of the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations

This paper is focused on the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations using path integration based on the Gauss-Legendre integration scheme. First applying methods of harmonic balance and multiple scales to the deterministic case, the stabilities of the responses can be analyzed. Then the steady state periodic solution of probability density can be captured via path integration. At the same time, the changes of probability density induced by the intensities of harmonic and stochastic excitations are discussed in three cases.

Study of the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations by path integration

In this paper, the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations is investigated via path integration based on the Gauss-Legendre integration formula. The method can successfully capture the steady state periodic solution of probability density function. This path integration method, using the periodicity of the coefficient of associated Fokker-Planck-Kolmogorov equation, is extended to deal with the averaged stationary probability density, and is efficient to computation. Meanwhile, the changes of probability density caused by the intensities of harmonic and stochastic excitations, are discussed in three cases through the instantaneous probability density and the averaged stationary probability density.

Computations of shallow water equations with high-order central-upwind schemes on triangular meshes

New high-order central-upwind schemes on triangular meshes are proposed to approximate the solutions of shallow water equations. The nonuniform width of the different local Riemann fans is calculated more accurately, and the new central-upwind schemes are of simplicity, universality and robustness. At the same time, due to the central-upwind schemes are combined with new reconstructions based on adaptive least squares, the schemes have the almost non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our methods.

Numerical simulation for two-phase flows using hybrid scheme

The present paper is devoted to the computation of two-phase flows using two-fluid approach. Two difficulties arise for the model: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. To deal with the aforementioned difficulties, the non-conservative volume fraction equation is discretized by WENO scheme, and the CWENO-type central-upwind scheme is applied to solve the mass, momentum and energy equations. Computational results are eventually provided and discussed.

A HYBRID SCHEME FOR THREE-DIMENSIONAL INCOMPRESSIBLE TWO-PHASE FLOWS

We present a hybrid scheme for computations of three-dimensional incompressible two-phase flows. A Poisson-like pressure equation is deduced from the incompressible constraint, i.e., the divergence-free condition of the velocity field, via an extended marker and cell method, and the moment equations in the 3D incompressible Navier–Stokes equations are solved by our 3D semi-discrete Hermite central-upwind scheme. The interface between the two fluids is considered to be the 0.5 level set of a smooth function being a smeared out Heaviside function. Numerical results are offered to verify the desired efficiency and accuracy of our 3D hybrid scheme.

Computations of steady and unsteady transport of pollutant in shallow water

We present new models for simulating the steady and unsteady transport of pollutant. Then the simple central-upwind schemes based on central weighted essentially non-oscillatory reconstructions are proposed in this paper for computing the one- and two-dimensional steady and unsteady models. Since the non-uniform width of the different local Riemann fans is calculated more accurately, the central-upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided. Synchronously, due to the central-upwind schemes are combined with the fourth-order central weighted essentially non-oscillatory reconstructions, the schemes have the non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our methods.

NUMERICAL SIMULATION OF 2D LIQUID SLOSHING

In this paper, we present an extended ghost fluid method (GFM) for computations of liquid sloshing in incompressible multifluids consisting of inviscid and viscous regions. That is, the sloshing interface between inviscid and viscous fluids is tracked by the zero contour of a level set function and the appropriate sloshing interface conditions are captured by defining ghost fluids that have the velocities and pressure of the real fluid at each point while fixing the density and the kinematic viscosity of the other fluid. Meanwhile, a second order single-fluid solver, the central-weighted-essentially-nonoscillatory(CWENO)-type central-upwind scheme, is developed from our previous works. The high resolution and the nonoscillatory quality of the scheme can be verified by solving several numerical experiments. Nonlinear sloshing inside a pitching partially filled rectangular tank with/without baffles has been investigated.

Numerical meshfree path integration method for non-linear dynamic systems

Abstract We present a new numerical meshfree path integration (MPI) method for non-linear dynamic systems. The obtained MPI method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the MPI method and the ordinary differential equations for the first and second-order moments on the basis of Gaussian closure method. The piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions. The good performance of the resulting method is finally shown in the numerical examples by using three specific non-linear dynamic systems: Duffing oscillator subjected to both harmonic and stochastic excitations, Duffing–Rayleigh oscillator subjected to both harmonic and stochastic excitations, and CHEN system driven by three different Gaussian white noises.

FLUID SIMULATION USING AN ADAPTIVE SEMI-DISCRETE CENTRAL-UPWIND SCHEME

In this paper, we propose an adaptive semi-discrete central-upwind scheme on triangular meshes to approximate the solutions of hyperbolic conservation laws in gas dynamics: a local smoothness indicator is implemented to identify discontinuous solution regions; away from the discontinuities, a simple and computationally economical flux is used; near the discontinuities, an accurate but computationally uneconomical flux is used. Several numerical experiments, Euler equations and Kelvin–Helmholtz instability, are successfully simulated. The numerical results show the desired accuracy, high efficiency and robustness of the adaptive scheme.