arXiv subjects

Changsheng Yu

Publications and source records attributed to Changsheng Yu.

8 recordsLinked to original sources

arXiv2026

Selection criteria for the compressible Euler equations: numerical study

We numerically investigate selection criteria for dissipative weak solutions of the compressible Euler equations. Using the Kelvin-Helmholtz problem as a benchmark, we compare four standard numerical schemes - the viscous finite volume (VFV) method, the discrete velocity Boltzmann finite volume (DVBFV) method, the Runge-Kutta discontinuous Galerkin (RKDG) method, and the discontinuous Galerkin spectral element method (DGSEM), and demonstrate that each may converge to a different dissipative weak solution. We evaluate these solutions with respect to several selection criteria based on entropy production, total energy, and energy defect, and examine the sensitivity of selection functionals to the numerical diffusion inherent in each scheme.

math-ph↗

arXiv2026

On admissible solutions to the coupled Riemann problem with heat-flux discontinuity

We study the Riemann problem for the compressible Euler equations with a stationary coupling interface across which a discontinuity in the heat flux is prescribed. This coupling gives rise to non-conservative effects and models heat addition mechanisms such as condensation-induced waves. Without imposing restrictions on sonic states, we analyze the problem in all Mach number regimes. Lax weak entropy solutions are constructed via half-Riemann problems, and we show that non-uniqueness occurs for a large class of initial data. To address this, we introduce an admissibility criterion derived from the evolutionarity criterion, and we characterize the full structure of admissible Riemann solutions. Our analysis establishes local existence of admissible Riemann solutions provided the heat flux jump is sufficiently small, while also identifying families of initial data for which admissible Riemann solutions cannot exist for any fixed, nonzero heat flux jump. Numerical experiments are included to illustrate the theoretical findings.

math.AP↗

arXiv2026

Numerical Study of Dissipative Weak Solutions for the Euler Equations of Gas Dynamics

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation central-upwind (LDCU), and viscous finite volume (VFV) methods, whose higher-order extensions are obtained via the framework of the alternative weighted essentially non-oscillatory (A-WENO) schemes. These methods are applied to several benchmark problems, including several two-dimensional Riemann problems and a Kelvin-Helmholtz instability test. The numerical results demonstrate that for methods converging only weakly in space and time, the limiting solutions are generalized DW solutions, approximated in the sense of ${\cal K}$-convergence and dependent on the numerical scheme. For all of the studied methods, we compute the associated Young measures and compare the DW solutions using entropy production and energy defect criteria.

math.NA↗

arXiv2025

Oscillatory approximations and maximum entropy principle for the Euler system of gas dynamics

We show that the measure-valued solutions of the Euler system of gas dynamics generated by oscillatory sequences of consistent approximations violate the principle of maximal entropy production formulated by Dafermos. Numerical results illustrate that solutions obtained by standard numerical methods may be oscillatory and thus do not comply with the Dafermos criterion.

math.AP↗

arXiv2024

Monte Carlo method and the random isentropic Euler system

We show several results on convergence of the Monte Carlo method applied to consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our method is based on combination of several new concepts. We work with the dissipative weak solutions that can be seen as a universal closure of consistent approximations. Further, we apply the set-valued version of the Strong law of large numbers for general multivalued mapping with closed range and the Koml\'os theorem on strong converge of empirical averages of integrable functions. Theoretical results are illustrated by a series of numerical simulations obtained by an unconditionally convergent viscosity finite volume method combined with the Monte Carlo method.

math.NA↗

arXiv2022

A well-balanced scheme for Euler equations with singular sources

Numerical methods for the Euler equations with a singular source are discussed in this paper. The stationary discontinuity induced by the singular source and its coupling with the convection of fluid presents challenges to numerical methods. We show that the splitting scheme is not well-balanced and leads to incorrect results; in addition, some popular well-balanced schemes also give incorrect solutions in extreme cases due to the singularity of source. To fix such difficulties, we propose a solution-structure based approximate Riemann solver, in which the structure of Riemann solution is first predicted and then its corresponding approximate solver is given. The proposed solver can be applied to the calculation of numerical fluxes in a general finite volume method, which can lead to a new well-balanced scheme. Numerical tests show that the discontinuous Galerkin method based on the present approximate Riemann solver has the ability to capture each wave accurately.

math.NA↗

arXiv2022

Riemann problem of Euler equations with singular sources

This paper is concerned with the Riemann problem of one-dimensional Euler equations with a singular source. The exact solution of this Riemann problem contains a stationary discontinuity induced by the singular source, which is different from all the simple waves in the Riemann solution of classical Euler equations. We propose an eigenvalue-based monotonicity criterion to select the physical curve of this stationary discontinuity. By including this stationary discontinuity as an elementary wave, the structure of Riemann solution becomes diverse, e.g. the number of waves is not fixed and interactions between two waves become possible. Under the double CRP framework, we prove all possible structures of the Riemann solution.

math.AP↗

arXiv2022

Riemann problem for constant flow with single-point heating source

This work focuses on the Riemann problem of Euler equations with global constant initial conditions and a single-point heating source, which comes from the physical problem of heating one-dimensional inviscid compressible constant flow. In order to deal with the source of Dirac delta-function, we propose an analytical frame of double classic Riemann problems(CRPs) coupling, which treats the fluids on both sides of the heating point as two separate Riemann problems and then couples them. Under the double CRPs frame, the solution is self-similar, and only three types of solution are found. The theoretical analysis is also supported by the numerical simulation. Furthermore, the uniqueness of the Riemann solution is established with some restrictions on the Mach number of the initial condition.

math.AP↗