arXiv subjects

Shipeng Mao

Publications and source records attributed to Shipeng Mao.

5 recordsLinked to original sources

arXiv2026

Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system

In this paper, we propose and analyze a class of high-order numerical schemes within a fully discrete finite element framework for the incompressible Navier-Stokes equations with no-slip boundary conditions. The temporal discretization employs a kth-order (k=1,...,6) implicit-explicit backward difference formula (IMEX-BDFk), in which the nonlinear convection term is treated explicitly and the linear Stokes part implicitly, whereas the spatial discretization utilizes Taylor-Hood finite elements. We establish the stability and uniform boundedness of the numerical solution. We further establish optimal order error estimates in both space and time without any CFL-type condition, in the sense that the time step is independent of the spatial mesh size. In three dimensions, these include L2- and H1-norm error estimates for the velocity and L2-norm error estimates for the pressure, with temporal convergence rates up to sixth order for all variables. Numerical experiments are presented to demonstrate the effectiveness of the scheme and to confirm the theoretical convergence rates.

math.NA↗

arXiv2026

Stabilization-free virtual element methods based on finite element interpolation

In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator from the virtual element space to a (local) finite element space. Leveraging the properties of this operator, we design two types of stabilization-free schemes. The first scheme requires the interpolation to preserve the polynomial consistency related to the bilinear forms, thereby maintaining both consistency and stability as in the standard VEM. The second scheme relaxes this consistency requirement. While it may not satisfy the standard polynomial consistency, the second scheme retains optimal convergence with simpler construction, fewer degrees of freedom and, more importantly, applicable to more complex problems such as those involving nonlinearities or variable coefficients. We construct concrete interpolation operators for both conforming and nonconforming virtual elements in two and three dimensions. These operators are then employed to realize stabilization-free schemes for conforming and nonconforming VEMs. Numerical experiments confirm the optimal convergence rates of the proposed methods. The presented framework can be extended to design stabilization-free schemes for other polytopal discretization methods, such as the hybrid high-order method and the weak Galerkin method.

math.NA↗

arXiv2026

A Second-Order Maximum-Principle-Preserving Crouzeix-Raviart Finite Element Method for Time-dependent Transport Equation

In this paper, we construct an explicit, second-order, and maximum-principle-preserving Crouzeix-Raviart (CR) finite element method for two-dimensional time-dependent transport equation. The key observation is that the mass matrix of the CR element is with diagonal structure, which allows us to avoid the need to solve a large linear system for each time step and help to construct a low-order scheme that preserves the maximum principle in a simple way. We first introduce low-order schemes based on minimum and bilinear viscosities, and then recover second-order accuracy by means of greedy and flux-corrected transport viscosities. For inflow boundary conditions, we further design a modified FCT limiter. In addition, we propose a simple reconstruction based on Wachspress coordinates to obtain a continuous piecewise linear approximation on the $\frac{h}2$-mesh that satisfies the maximum principle on the whole domain. Under divergence-free velocity fields, the proposed schemes are conservative. Numerical experiments on both smooth and discontinuous test cases, with both solenoidal and non-solenoidal velocity fields, confirm the accuracy and robustness of the proposed schemes.

math.NA↗

arXiv2023

Lyapunov exponents and Lagrangian chaos suppression in compressible homogeneous isotropic turbulence

We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach number $M_t$ and Taylor-scale Reynolds number $Re_λ$. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as in incompressible flow due to density-velocity coupling. Modulus of the smallest Lyapunov exponent $λ_3$ provides the principal Lyapunov exponent of the time-reversed flow, which usually is wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent $λ_1$, determines all the exponents due to the vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier-Stokes equations and tracking particles verify these theoretical predictions. We found that: 1) The largest normalized Lyapunov exponent $λ_1 τ_η$, where $τ_η$ is the Kolmogorov time scale, is a decreasing function of $M_t$. Its dependence on $Re_λ$ is weak when the driving force is solenoidal, while it is an increasing function of $Re_λ$ when the solenoidal and compressible forces are comparable. Similar facts hold for $|λ_3|$, in contrast with well-studied short-correlated model; 2) The ratio of the first two Lyapunov exponents $λ_1/λ_2$ decreases with $Re_λ$, and is virtually independent of $M_t$ for $M_t \le 1$ in the case of solenoidal force but decreases as $M_t$ increases when solenoidal and compressible forces are comparable; 3) For purely solenoidal force, $λ_1 :λ_2 :λ_3 \approx 4:1:-5$ for $Re_λ> 80$, which is consistent with incompressible turbulence studies; 4) The ratio of dilation-to-vorticity is a more suitable parameter to characterize LEs than $M_t$.

physics.flu-dyn↗

arXiv2023

MHDnet: Physics-preserving learning for solving magnetohydrodynamics problems

Designing efficient and high-accuracy numerical methods for complex dynamic incompressible magnetohydrodynamics (MHD) equations remains a challenging problem in various analysis and design tasks. This is mainly due to the nonlinear coupling of the magnetic and velocity fields occurring with convection and Lorentz forces, and multiple physical constraints, which will lead to the limitations of numerical computation. In this paper, we develop the MHDnet as a physics-preserving learning approach to solve MHD problems, where three different mathematical formulations are considered and named $B$ formulation, $A_1$ formulation, and $A_2$ formulation. Then the formulations are embedded into the MHDnet that can preserve the underlying physical properties and divergence-free condition. Moreover, MHDnet is designed by the multi-modes feature merging with multiscale neural network architecture, which can accelerate the convergence of the neural networks (NN) by alleviating the interaction of magnetic fluid coupling across different frequency modes. Furthermore, the pressure fields of three formulations, as the hidden state, can be obtained without extra data and computational cost. Several numerical experiments are presented to demonstrate the performance of the proposed MHDnet compared with different NN architectures and numerical formulations.

math.NA↗