arXiv subjects

Jiequan Li

Publications and source records attributed to Jiequan Li.

25 records · Page 2Linked to original sources

arXiv2017

A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws

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.

math.NA↗

arXiv2017

A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows

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.

physics.flu-dyn↗

arXiv2017

Thermodynamical Effects and High Resolution Methods for Compressible Fluid Flows

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.

math.NA↗

arXiv2016

A Few Benchmark Test Cases for Higher-order Euler Solvers

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.

math.NA↗

arXiv2016

An Efficient and Accurate Two-Stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations

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.

math.NA↗

arXiv2015

A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers. I. Hyperbolic Conservation Laws

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.

math.NA↗

arXiv2013

The generalized Riemann problems for hyperbolic balance laws: A unified formulation towards high order

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.

math.NA↗