Jiequan Li

Liang Pan, Kun Xu, Qibing Li et al.

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 short) time stepping method has been recently proposed in the design of a fourth-order time accurate method [18]. In this paper, based on the same time-stepping method and the second-order GKS flux function [34], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [21], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme, even though the third- and fourth-order schemes have similar computation cost. Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. Perfect numerical solutions can be obtained from the high Reynolds number boundary layer solutions to the hypersonic viscous heat conducting flow computations. Many numerical tests, including many difficult ones for the Navier-Stokes solvers, have been used to validate the current fourth-order method. Following the two-stage time-stepping framework, the one-stage third-order GKS can be easily extended to a fifth-order method with the usage of both first-order and second-order time derivatives of the flux function. The use of time-accurate flux function may have great impact on the development of higher-order CFD methods.

Jiequan Li, Zhifang Du

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 (L-W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using well-developed four stages for R-K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L-W type schemes for the construction of a two-stage fourth order scheme.

Dinshaw S. Balsara, Jiequan Li, Gino I. Montecinos

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 Riemann problem is different for each hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that is universal in that it is applicable to any hyperbolic system, whether in conservation form or with non-conservative products. The HLLI Riemann solver is also complete in the sense that if it is given a complete set of eigenvectors, it represents all waves with minimal dissipation. It is, therefore, very attractive to build a generalized Riemann problem solver version of the HLLI Riemann solver. This is the task that is accomplished in the present paper. We show that at second order, the generalized Riemann problem version of the HLLI Riemann solver is easy to design. Our GRP solver is also complete and universal because it inherits those good properties from original HLLI Riemann solver. We also show how our GRP solver can be adapted to the solution of hyperbolic systems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performs robustly and well over a range of test problems. All implementation-related details are presented. Results from several stringent test problems are shown. These test problems are drawn from many different hyperbolic systems, and include hyperbolic systems in conservation form; with non-conservative products; and with stiff source terms. The present generalized Riemann problem solver performs well on all of them.

Zhifang Du, Jiequan Li

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 [{\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. Sci. Comput., 38 (2016), pp.~A3046--A3069}]. Instead of computing the first moment of the solution additionally in the conventional HWENO or DG approach, we can directly take the {\em interface values}, which are already available in the numerical flux construction using the generalized Riemann problem (GRP) solver, to approximate the first moment. The resulting scheme is fourth order temporal accurate by only invoking the HWENO reconstruction twice so that it becomes more compact. Numerical experiments show that such compactness makes significant impact on the resolution of nonlinear waves.

Zhifang Du, Jiequan Li

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. Sci. Comput., 38 (2016), pp.~A3046--A3069}]. As such, it is significant when capturing small scale structures near physical boundaries. Different from previous contributions in literature, the current approach constructs a fourth order accurate approximation to boundary conditions by only using the Jacobian of the flux function (characteristic information) instead of its successive differentiation leading to tensors of high ranks in the inverse Lax-Wendroff method. Technically, data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells, and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two-stage scheme over boundary cells fully consistent with that over interior cells. This highlights the fact that {\em continuous boundary conditions only match continuous partial differential equations (PDEs), and they must be approximated in a consistent way (even though it could be exactly valued) when the PDEs are discretized.} Several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions.

Jiequan Li

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 block is the Riemann solution labeled as the solution element "1". Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as "$1\odot 1\odot 1\odot 1=4$". The other is the one-stage Lax-Wendroff (L-W) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics, labeled as "$2$", are taken as the building black. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as "$2\odot 2=4$", are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the "$2\odot 2=4$" algorithm extracts the advantages of the method of line and one-stage L-W method. As a core part, the pair "$2$" is expounded and L-W solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of L-W flow solver, and the gas kinetic scheme (GKS) solver, the microscopic L-W solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented.

Jiequan Li, Yue Wang

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 processes and verified for some first order schemes. In this paper, we refine the GRP solver to illustrate how the thermodynamical variation is integrated into the design of high resolution methods for compressible fluid flows and demonstrate numerically the importance of thermodynamic effect in the resolution of strong waves. As a by-product, we show that the GRP solver works for generic equations of state, and is independent of technical arguments.

Xin Lei, Jiequan Li

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 flows. Then we construct a conservative Godunov-based scheme with the high order accurate extension by using the generalized Riemann problem (GRP) solver, through the detailed analysis of kinetic energy exchange when fluids are mixed under the hypothesis of isobaric equilibrium. Numerical experiments are carried out for the shock-interface interaction and shock-bubble interaction problems, which display the excellent performance of this type of schemes and demonstrate that nonphysical oscillations are suppressed around material interfaces substantially.

Jianzhen Qian, Jiequan Li, Shuanghu Wang

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 primary building block in the construction of many numerical schemes (most notably the Godunov scheme), the analytic study of GRP will lead to an array of GRP schemes, which extend the Godunov scheme. Currently there are extensive studies on the second-order GRP scheme, which proves to be robust and is capable of resolving complex multidimensional fluid dynamic problems [M. Ben-Artzi and J. Falcovitz, "Generalized Riemann Problems in Computational Fluid Dynamics", Cambridge University Press, 2003]. A more general formulation of the second-order GRP solver is still confined with a class of weakly coupled systems [Numer. Math. (2007) 106:369-425]. This paper provides a unified approach for solving the GRP in the general context of hyperbolic balance laws, without weakly coupled constraint, towards high order accuracy. The derivation of the second-order GRP solver is more concise compared to those in previous works and the third-order quadratic GRP is resolved for the first time. The latter is shown to be necessary through numerical experiments with strong discontinuities. Our method relies heavily on the new treatment of the rarefaction wave by deriving the L(Q)-equations, an ODE system capturing the "evolution" of the characteristic derivatives in x-t space for generalized Riemann invariants. The case of a sonic point is incorporated into a general treatment. The accuracy of the derived GRP solvers are justified and numerical examples are presented for the performance of the resulting scheme.

Liang Pan, Jiequan Li, Kun Xu

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 testing higher-order schemes. As reference solutions, all test cases will be calculated by our recently developed two-stage fourth-order gas-kinetic scheme (GKS). All examples are selected so that the numerical settings are very simple and any high order accurate scheme can be straightly used for these test cases, and compare their performance with the GKS solutions. The examples include highly oscillatory solutions and the large density ratio problem in one dimensional case; hurricane-like solutions, interactions of planar contact discontinuities (the composite of entropy wave and vortex sheets) sheets with large Mach number asymptotic, interaction of planar rarefaction waves with transition from continuous flows to the presence of shocks, and other types of interactions of two-dimensional planar waves. The numerical results from the fourth-order gas-kinetic scheme provide reference solutions only. These benchmark test cases will help CFD developers to validate and further develop their schemes to a higher level of accuracy and robustness.