# Jiequan LiCapital Normal University · Academy of Multidisciplinary Studies

Jiequan Li

Professor

## About

116

Publications

21,883

Reads

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2,344

Citations

Citations since 2017

Introduction

---PDE theories related to flow structures, e.g.,2-D Riemann problems in gas dynamics;
---Fundamental principles of finite volume methods and numerical analysis;
---High order CFD algorithms, particularly multi-stage temporal-spatial coupling methods based on Lax-Wendroff type flow solvers, e.g., GRP solvers;
---Applications to multi-material/phase problems and turbulences.

Additional affiliations

August 2015 - September 2022

December 2010 - July 2015

August 2002 - December 2010

**Capital Normal Univesity**

Position

- Professor of Mathematics

Education

September 1994 - December 1996

September 1991 - July 1994

**Beijing Normal University**

Field of study

- Mathematics

## Publications

Publications (116)

It was generally expected that monotone schemes are oscillation-free for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the high...

In this paper we develop a novel two-stage fourth order time-accurate
discretization for time-dependent flow problems, particularly for hyperbolic
conservation laws. Different from the classical Runge-Kutta (R-K) temporal
discretization for first order Riemann solvers as building blocks, the current
approach is solely associated with Lax-Wendroff (...

Resumen Resumen
We investigate<sup> </sup>the problem of two-dimensional, unsteady expansion of an inviscid , polytropic <sup> </sup>gas, which can be interpreted as the collapse of a<sup> </sup>wedge-shaped dam containing water initially with a uniform velocity. We<sup> </sup>model this problem by isentropic Euler equations. The flow is<sup> </su...

This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume a...

This paper proposes a generalized Riemann problem (GRP)-based high resolution ghost fluid method (GFM) for the simulation of 1-D multi-medium compressible fluid flows. A kind of linearly distributed ghost fluid states is defined via a local double-medium generalized Riemann problem (GRP) at the material interface. The advantages of the GRP-based GF...

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the...

As an extension of the two-stage fourth-order subcell finite volume (SCFV) method that we developed for two-dimensional (2D) compressible flows, this study continues our efforts toward three-dimensional (3D) simulations on hexahedral meshes. The two components of subcell divisions and two-stage fourth-order time stepping are utilized to improve eff...

As an extension of the two-stage fourth-order subcell finite volume (SCFV) method that we developed for two-dimensional (2D) compressible flows 1 , this study continues our efforts toward three-dimensional (3D) simulations on hexahedral meshes. The two components of subcell divisions and two-stage fourth-order time stepping are utilized to improve...

In the computation of compressible fluid flows, numerical boundary conditions are always necessary for all physical variables at computational boundaries while just partial physical variables are often prescribed as physical boundary conditions. Certain extrapolation technique or ghost cells are often employed traditionally for this issue but spuri...

The volume of fluid (VOF) method is a successful approach to track the dynamics of free boundaries thanks to its flexibility, efficiency and simplicity. Since its advent there are various efficient numerical methods to solve the associated VOF equation. This paper applies the two-stage fourth order accurate time discretization in [13, SIAM J. Sci....

This chapter deals with multi-material flow problems by a kind of effective numerical methods, based on a series of reduced forms of the Baer-Nunziato (BN)-type model. Numerical simulations often face a host of difficult challenges, typically including the volume fraction positivity and stability of multi-material shocks. To cope with these challen...

To meet the demand for complex geometries and high resolutions of small-scale flow structures, a two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques for compressible flows on (unstructured) triangular meshes was developed to improve the compactness and efficiency. Compared to the...

An efficient gas-kinetic scheme with fourth-order accuracy in both space and time is developed for the Navier-Stokes equations on triangular meshes. The scheme combines an efficient correction procedure via reconstruction (CPR) framework with a robust gas-kinetic flux formula, which computes both the flux and its time-derivative. The availability o...

The kinetic theory provides a physical basis for developing multiscal methods for gas flows covering a wide range of flow regimes. A particular challenge for kinetic schemes is whether they can capture the correct hydrodynamic behaviors of the system in the continuum regime (i.e., as the Knudsen number ǫ ≪ 1) without enforcing kinetic scale resolut...

To meet the demand for complex geometries and high resolutions of small-scale flow structures, a two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques for compressible flows over (unstructured) triangular meshes was developed to improve the compactness and efficiency. Compared to t...

The equation of state (EOS) embodies thermodynamic properties of compress-ible fluid materials and usually has very complicated forms in real engineering applications, subject to the physical requirements of thermodynamics. The complexity of EOS in form gives rise to the difficulty in analyzing relevant wave patterns. Concerning the design of numer...

In the computation of compressible fluid flows, numerical boundary conditions are 4 always necessary for all physical variables at computational boundaries while just partial physical 5 variables are often prescribed as physical boundary conditions. Certain extrapolation technique or 6 ghost cells are often employed traditionally for this issue but...

When describing the deflagration-to-detonation transition in solid granular explosives mixed with gaseous products of combustion, a well-developed two-phase mixture model is the compressible Baer-Nunziato (BN) model of flows containing solid and gas phases. As this model is numerically simulated by a conservative Godunov-type scheme, spurious oscil...

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity...

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity...

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decomposi-tions, we establish the global existence and regularit...

A highly efficient gas-kinetic scheme with fourth-order accuracy in both space and time is developed for the Navier-Stokes equations on triangular meshes. The scheme combines an efficient correction procedure via reconstruction (CPR) framework with a robust gas-kinetic flux formula, which computes both the flux and its time-derivative. The availabi...

This chapter deals with multi-material flow problems by a kind of effective numerical methods, based on a series of reduced forms of the Baer-Nunziato (BN) model. Numerical simulations often face a host of difficult challenges, typically including the volume fraction positivity and stability of multi-material shocks. To cope with these challenges,...

A spacetime outlook on Computational Fluid Dynamics is advocated: models in
fluid mechanics often have the spacetime correlation property, which should be
inherited and preserved in the corresponding numerical algorithms. Starting
from the fundamental formulation of fluid mechanics under continuum hypothesis,
this paper defines the meaning of space...

When describing the deflagration-to-detonation transition in solid granular explosives mixed with gaseous products of combustion, a well-developed two-phase mixture model is the compressible Baer-Nunziato (BN) model, containing solid and gas phases. If this model is numerically simulated by a conservative Godunov-type scheme, spurious oscillations...

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the ``meaningful objects'' are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in t...

In this paper, we present a new two-stage fourth-order ﬁnite difference weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with special application to compressible Euler equations. To construct this algorithm, apart from the traditional WCNS for the spatial derivative, it was necessary to ﬁrst construct a linear compact/expli...

A The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are...

This is my talk slide about the thinking of numerical simulations.

This paper focuses on the structure of classical sonic-supersonic solutions near sonic curves for the two-dimensional full Euler equations in gas dynamics. In order to deal with the parabolic degeneracy near the sonic curve, a novel set of dependent and independent variables are introduced to transform the Euler equations into a new system of gover...

This paper focuses on the structure of classical sonic-supersonic solutions near sonic curves for the two-dimensional full Euler equations in gas dynamics. In order to deal with the parabolic degeneracy near the sonic curve, a novel set of dependent and independent variables are introduced to transform the Euler equations into a new system of gover...

The kinetic theory provides a physical basis for developing multiscal methods for gas flows covering a wide range of flow regimes. A particular challenge for kinetic schemes is whether they can capture the correct hydrodynamic behaviors of the system in the continuum regime (i.e., as the Knudsen number $\epsilon\ll 1$ ) without enforcing kinetic sc...

The simulation of compressible multi-fluid flows receives increasing attention, thanks to extensive engineering applications, but there are still many bottleneck problems remain unresolved due to the presence of singularities (shocks, material interfaces, vortices and other discontinuities etc) in flows, which arises notorious diffculties in all as...

Reliable tracking of moving boundaries is important for the simulation of compressible fluid flows and there are a lot of contributions in literature. We recognize from the classical piston problem, a typical moving boundary problem in gas dynamics, that the acceleration is a key element in the description of the motion and it should be incorporate...

Reliable tracking of moving boundaries is important for the simulation of compressible fluid flows and there are a lot of contributions in literature. We recognize from the classical piston problem, a typical moving boundary problem in gas dynamics, that the acceleration is a key element in the description of the motion and it should be incorporate...

Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimension...

We develop a new fourth-order discontinuous Galerkin method using the generalized Riemann problem (GRP) solver based on the framework of the two-stage fourth-order accurate temporal discretization, with special application to compressible Euler equations. The appealing advantage of the two-stage fourth-order accurate temporal discretization is that...

This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of "balance laws". Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume a...

This paper addresses the three concepts of \textit{ consistency, stability and convergence } in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of ``balance laws''. Such laws express the relevant physical conservation laws in the presence of discontinuities. Fi...

This paper serves to simulate compressible multi-fluid flows using an energy-splitting method based on the generalized Riemann problem (GRP) solver. The emphasis is put on two-dimensional problems to address the fundamental role of transversal effect in the construction of numerical fluxes, which exhibits the genuine multi-dimensionality of the und...

With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct “physics”. There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building bloc...

Finite volume schemes for the two-dimensional (2D) wave system are taken to demonstrate the role of the genuine dimensionality of Lax-Wendroff flow solvers for compressible fluid flows. When the finite volume schemes are applied, the transversal variation relative to the computational cell interfaces is neglected, and only the normal numerical flux...

With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct "physics". There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building bloc...

The Riemann problem, and the associated generalized Riemann problem, are increasingly seen as the important building blocks for modern higher order Godunov-type schemes. In the past, building a generalized Riemann problem solver was seen as an intricately mathematical task for complicated physical or engineering problems because the associated Riem...

This paper serves to treat boundary conditions numerically with high order accuracy in order to match the two-stage fourth-order finite volume schemes for hyperbolic problems developed in [{\em J. Li and Z. Du, A two-stage fourth order time-accurate discretization {L}ax--{W}endroff type flow solvers, {I}. {H}yperbolic conservation laws, SIAM, J. Sc...

This paper proposes a new non-oscillatory {\em energy-splitting} conservative algorithm for computing multi-fluid flows in the Eulerian framework. In comparison with existing multi-fluid algorithms in literatures, it is shown that the mass fraction model with isobaric hypothesis is a plausible choice for designing numerical methods for multi-fluid...

This paper develops a new fifth order accurate Hermite WENO (HWENO) reconstruction method for hyperbolic conservation schemes in the framework of the two-stage fourth order accurate temporal discretization in Li and Du (2016) [13]. Instead of computing the first moment of the solution additionally in the conventional HWENO or DG approach, we can di...

One of the fundamental differences of compressible fluid flows from incompressible fluid flows is the involvement of thermodynamics. This difference should be manifested in the design of numerical methods and seems often be neglected in addition that the entropy inequality, as a conceptual derivative, is taken into account to reflect irreversible p...

This paper establishes the entropy convergence of a new two-value high resolution finite volume scheme with slope relaxation for conservation laws. This scheme, motivated by the general method of high resolution schemes that have high-order accuracy in smooth regions of solutions and are free of oscillations near discontinuities, unifies and evolve...

There have been great efforts on the development of higher-order numerical schemes for compressible Euler equations. The traditional tests mostly targeting on the strong shock interactions alone may not be adequate to test the performance of higher-order schemes. This study will introduce a few test cases with a wide range of wave structures for te...

For computational fluid dynamics (CFD), the generalized Riemann
problem (GRP) solver and the second-order gas-kinetic scheme (GKS)
provide a time-accurate flux function starting from a discontinuous
piecewise linear flow distributions around each cell interface. With
the use of time derivative of the flux function, a two-stage
Lax-Wendroff-type (L-...

In this paper, a fully discrete high-resolution ALE method is developed over untwisted time-space control volumes. In the framework of the finite volume method, 2D Euler equations are discretized over untwisted moving control volumes, and the resulting numerical flux is computed using the generalized Riemann problem (GRP) solver. Then, the fluid fl...

For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the gas-kinetic kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around each cell interface. With the use of time derivative of the flux function, a two-stage Lax-Wendroff-type (L-W for...

In this paper, a remapping-free adaptive GRP method for one dimensional (1-D) compressible flows is developed. Based on the framework of finite volume method, the 1-D Euler equations are discretized on moving volumes and the resulting numerical fluxes are computed directly by the GRP method. Thus the remapping process in the earlier adaptive GRP al...

The generalized Riemann problems (GRP) for nonlinear hyperbolic systems of balance laws in one space dimension are now well-known and can be formulated as follows: Given initial data which are smooth on two sides of a discontinuity, determine the time evolution of the solution near the discontinuity. In particular, the GRP of (k+1)(k+1)th order hig...

The Harten-Lax-van Leer scheme is popularly used in the CFD community. However, oscillations are observed from shock tube problems for stiffened gases when the adiabatic index is greater than 3. To understand this phenomenon, the dissipation effect of the scheme is evaluated quantitatively in terms of dissipation matrices. We have proven and numeri...

The von Neumann (discrete Fourier) analysis and modified equation technique have been proven to be two effective tools in the design and analysis of finite difference schemes for linear and nonlinear problems. The former has merits of simplicity and intuition in practical applications, but only restricted to problems of linear equations with consta...

Unstructured mesh methods have attracted much attention in CFD community due to the flexibility for dealing with complex geometries and the ability to easily incorporate adaptive (moving) mesh strategies. When the finite volume framework is applied, a reliable solver is crucial for the construction of numerical fluxes, for which the generalized Rie...

The Generalized Riemann Problems (GRP) for nonlinear hyperbolic systems of
balance laws in one space dimension are now well-known and can be formulated as
follows: Given initial-data which are smooth on two sides of a discontinuity,
determine the time evolution of the solution near the discontinuity. While the
classical Riemann problem serves as a...

As a ladder step to study transonic problems, we investigate two families of degenerate Goursat-type boundary value problems arising from the two-dimensional pseudo-steady isothermal Euler equations. The first family is about the genuinely two-dimensional full expansion of gas into a vacuum with a wedge; the other is a semi-hyperbolic patch that st...

The adaptive generalized Riemann problem (GRP) scheme for 2-D compressible fluid flows has been proposed in [J. Comput. Phys., 229 (2010), 1448-1466] and it displays the capability in overcoming difficulties such as the start-up error for a single shock, and the numerical instability of the almost stationary shock. In this paper, we will provide th...

This paper is concerned with classical solutions to the interaction of two arbitrary planar rarefaction waves for the self-similar Euler equations in two space dimensions. We develop the direct approach, started in Chen and Zheng (in press) [3], to the problem to recover all the properties of the solutions obtained via the hodograph transformation...

Oscillations are ubiquitous in numerical solutions obtained by high order or even first order schemes for hyperbolic problems and are conventionally understood as the consequence of low dissipation effects of underlying numerical schemes. Earlier analysis was done mainly through the effective discrete Fourier analysis for linear problems or the mod...