arXiv subjects

Xianghai Zhou

Publications and source records attributed to Xianghai Zhou.

3 recordsLinked to original sources

arXiv2024

The virtual element method for the three dimensional inductionless magnetohydrodynamic model

This paper proposes a novel first-order and a novel second-order fully discrete virtual element schemes based on the scalar auxiliary variable method for the three dimensional inductionless magnetohydrodynamics problem. The backward Eular formula and the backward differential formula are used for the time discretization and two types conservation virtual element formulations are employed for spatial discretization. The main advantages include that the mass conservation in the velocity field and the charge conservation in the current density field are kept by taking characteristics of the virtual element method's discrete complex structures, the nonlinear term is handled explicitly by applying the scalar auxiliary variable method, the current density field is decoupled from the momentum equation, and the velocity field is decoupled from the Ohm's law. The unconditionally stable of the two fully discrete schemes are demonstrated. Finally, we present numerical experiment to verify the valid of the proposed schemes.

math.AP↗

arXiv2023

A full divergence-free of high order virtual finite element method to approximation of stationary inductionless magnetohydrodynamic equations on polygonal meshes

In this present paper we consider a full divergence-free of high order virtual finite element algorithm to approximate the stationary inductionless magnetohydrodynamic model on polygonal meshes. More precisely, we choice appropriate virtual spaces and necessary degrees of freedom for velocity and current density to guarantee that their final discrete formats are both pointwise divergence-free. Moreover, we hope to achieve higher approximation accuracy at higher "polynomial" orders k_{1} \geq 2, k_{2} \geq 1, while the full divergence-free property has always been satisfied. And then we processed rigorous error analysis to show that the proposed method is stable and convergent. Several numerical tests are presented, confirming the theoretical predictions.

math.AP↗

arXiv2023

Stability and error analysis of the SAV schemes for the inductionless MHD equations

In this paper, we consider numerical approximations for solving the inductionless magnetohydrodynamic (MHD) equations. By utilizing the scalar auxiliary variable (SAV) approach for dealing with the convective and coupling terms, we propose some first- and second-order schemes for this system. These schemes are linear, decoupled, unconditionally energy stable, and only require solving a sequence of differential equations with constant coefficients at each time step. We further derive a rigorous error analysis for the first-order scheme, establishing optimal convergence rates for the velocity, pressure, current density and electric potential in the two-dimensional case. Numerical examples are presented to verify the theoretical findings and show the performances of the schemes.

math.NA↗