arXiv subjects

Bangwei She

Publications and source records attributed to Bangwei She.

At least 19 recordsLinked to original sources

arXiv2026

Convergence analysis of a MAC scheme for the barotropic Euler system

We study a Marker-and-Cell (MAC) scheme for the barotropic Euler system. First, we apply the recently developed Lax-type convergence theorem to show the convergence of the MAC scheme to i) a dissipative weak solution unconditionally and ii) a strong solution as long as it exists. Second, We derive relative energy error estimates up to the lifespan of a strong solution, without assuming uniform boundedness of the numerical sequence. Additionally, assuming the boundedness of the numerical solutions, we obtain the optimal relative energy rate of 1, corresponding to a convergence rate of 1/2 for the numerical solutions. Finally, we corroborate our theoretical results by numerical experiments.

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arXiv2026

Lax convergence theorems and error estimates of a finite element method for the incompressible Euler system

In this paper, we present convergence theorems for numerical solutions of the incompressible Euler equations. The first result is the Lax-Wendroff-type theorem, while the second can be formulated in the framework of the Lax equivalence theorem. To illustrate their application, we study a finite element method that uses a pair of $RT_0/P_0$ elements to approximate the velocity and pressure, respectively. Applying the concept of the relative energy, we derive the convergence rates of our numerical method using two different approaches. Finally, we validate the theoretical convergence results through numerical experiments.

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arXiv2026

Temperature-driven turbulence in compressible fluid flows

We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences 2, 2160 (2020)].

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arXiv2026

Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system

We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.

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arXiv2026

Stability and error estimates of a linear and partitioned finite element method approximating nonlinear fluid-structure interactions

We propose and analyze a linear and partitioned finite element method for fluid-shell interactions under the arbitrary Lagrangian-Eulerian (ALE) framework. We adopt the P1-bubble/P1/P1 elements for the fluid velocity, pressure, and structure velocity, respectively. We show the stability and error estimates of the scheme without assuming infinitesimal structural deformation nor neglecting fluid convection effects. The theoretical convergence rate is further corroborated by numerical experiments.

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arXiv2025

A multi-resolution limiter for the Runge-Kutta discontinuous Galerkin method

We propose a novel multi-resolution (MR) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws on a general unstructured mesh. Unlike classical limiters, which detects only solution discontinuities to dichotomize cells into good or troubled, the proposed MR limiter also takes into account the derivative discontinuities to divide cells into several groups. The method operates by performing a successive comparison of the local DG polynomial's derivatives, from high-order to low-order, against a baseline constructed from neighboring cell averages. If a $k$th-order derivative of the DG polynomial is larger than the baseline, then we reduce the order to $k-1$ and set the corresponding $k$th-order terms to be 0; Otherwise, the remaining $k$th-order DG polynomial is used to represent the final solution. Only if all the derivatives are larger than the baseline, a TVD slope limiter is used to reconstruct the solution. In this manner, the limiter dynamically selects an optimal polynomial suited to the local solution smoothness without problem-dependent parameter to tune. Notably, it also possesses a scale-invariance property that is absent in most classical limiters. A series of numerical examples demonstrate the accuracy and robustness of the proposed MR limiter.

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arXiv2024

Error analysis of the Monte Carlo method for compressible magnetohydrodynamics

We study random compressible viscous magnetohydrodynamic flows. Combining the Monte Carlo method with a deterministic finite volume method we solve the random system numerically. Quantitative error estimates including statistical and deterministic errors are analyzed up to a stopping time of the exact solution. On the life-span of an exact strong solution we prove the convergence of the numerical solutions. Numerical experiments illustrate rich dynamics of random viscous compressible magnetohydrodynamics.

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arXiv2024

What is a limit of structure-preserving numerical methods for compressible flows?

We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply K-convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect.

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arXiv2024

Convergence of numerical methods for the Navier-Stokes-Fourier system driven by uncertain initial/boundary data

We consider the Navier-Stokes-Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh-Benard convection problem.

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arXiv2023

Penalty method for the Navier-Stokes-Fourier system with Dirichlet boundary conditions: convergence and error estimates

We study the convergence and error estimates of a finite volume method for the compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions. Physical fluid domain is typically smooth and needs to be approximated by a polygonal computational domain. This leads to domain-related discretization errors, the so-called variational crimes. To treat them efficiently we embed the fluid domain into a large enough cubed domain, and propose a finite volume scheme for the corresponding domain-penalized problem. Under the assumption that the numerical density and temperature are uniformly bounded, we derive the ballistic energy inequality, yielding a priori estimates and the consistency of the penalization finite volume approximations. Further, we show that the numerical solutions converge weakly to a generalized, the so-called dissipative measure-valued, solution of the corresponding Dirichlet problem. If a strong solution exists, we prove that our numerical approximations converge strongly with the rate 1/4. Additionally, assuming uniform boundedness of the approximate velocities, we obtain global existence of the strong solution. In this case we prove that the numerical solutions converge strongly to the strong solution with the optimal rate 1/2.

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arXiv2023

Convergence analysis of the Monte Carlo method for random Navier--Stokes--Fourier system

In the present paper we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier--Stokes--Fourier system. The Monte Carlo method, which is frequently used for the approximation of statistical moments, is combined with a suitable deterministic discretisation method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to a statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for the Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results.

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arXiv2023

Stability and error estimates of a linear numerical scheme approximating nonlinear fluid-structure interactions

In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian-Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation. For time discretization, we employ the backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement. We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step $\tau$ and the mesh $h$. We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one, at least in stable configurations.

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arXiv2022

On the Motion of a Pendulum with a Cavity Filled with a Compressible Fluid

We study the motion of the coupled system, $\mathscr S$, constituted by a physical pendulum, $\mathscr B$, with an interior cavity entirely filled with a viscous, compressible fluid, $\mathscr F$. The presence of the fluid may strongly affect on the motion of $\mathscr B$. In fact, we prove that, under appropriate assumptions, the fluid acts as a damper, namely, $\mathscr S$ must eventually reach a rest-state. Such a state is characterized by a suitable time-independent density distribution of $\mathscr F$ and a corresponding equilibrium position of the center of mass of $\mathscr S$. These results are proved in the very general class of weak solutions and do not require any restriction on the initial data, other than having a finite energy. We complement our findings with some numerical tests. The latter show, among other things, the interesting property that ``large" compressibility favors the damping effect, since it drastically reduces the time that $\mathscr S$ takes to go to rest.

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arXiv2022

Error estimates of a finite volume method for the compressible Navier--Stokes--Fourier system

In this paper we study the convergence rate of a finite volume approximation of the compressible Navier--Stokes--Fourier system. To this end we first show the local existence of a highly regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.

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arXiv2022

Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system

In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we present an elegant alternative to a direct, but rather technical, analysis of variational crimes by means of the penalty approach. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

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arXiv2022

Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system

We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier-Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.

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arXiv2022

Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method

The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a finite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gy\"{o}ngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo finite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.

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arXiv2021

Error estimates of the Godunov method for the multidimensional compressible Euler system

We derive a priori error of the Godunov method for the multidimensional Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the $L^2$-norm of errors in density, momentum and entropy. Under the assumption that the numerical density and energy are bounded, we obtain a convergence rate of $1/2$ for the relative energy in the $L^1$-norm. Further, under the assumption -- the total variation of numerical solution is bounded, we obtain the first order convergence rate for the relative energy in the $L^1$-norm. Consequently, numerical solutions (density, momentum and entropy) converge in the $L^2$-norm with the convergence rate of $1/2$. The numerical results presented for Riemann problems are consistent with our theoretical analysis.

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