53 research outputs found
On a global supersonic-sonic patch characterized by 2-D steady full Euler equations
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-dimensional full system of
steady Euler equations and studies solution behaviors near sonic curves,
depending on the proper choice of boundary data extracted from the airfoil
problem and related contexts. New characteristic decompositions are developed
for the full system and a delicate local partial hodograph transformation is
introduced for the solution estimates. It is shown that the solution is
uniformly continuous up to the sonic curve and the sonic
curve is also continuous.Comment: 34 page
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
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.Comment: 25 pages, 12 figure
Consistency and Convergence of Finite Volume Approximations to Nonlinear Hyperbolic Balance Laws
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. Finite volume approximations employ this
viewpoint, and the present paper can be regarded as being in this category. It
is first shown that under very mild conditions a weak solution is indeed a
solution to the balance law. The schemes considered here allow the computation
of several quantities per mesh cell (e.g., slopes) and the notion of
consistency must be extended to this framework. Then a suitable convergence
theorem is established, generalizing the classical convergence theorem of Lax
and Wendroff. Finally, the limit functions are shown to be entropy solutions by
using a notion of "Godunov compatibility", which serves as a substitute to the
entropy condition.Comment: 27 page
A spacetime outlook on CFD: Spacetime correlated models and spacetime coupled algorithms
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 spacetime correlation of the models,
establishes the fundamental principle of finite volume schemes, expounds the
necessity of spacetime coupling of algorithms, as well as realizes the physical
and mathematical unification of basic governing equations of fluid mechanics
and finite volume schemes. In practice, the design methodology of spacetime
coupling high order numerical algorithms is presented, and the difference from
spacetime decoupling method is compared. It should be pointed out that most of
the contents in this paper are suitable for computational fluid dynamics under
the assumption of continuous medium, and some are only suitable for
compressible flow.Comment: in Chines
: Temporal-Spatial Coupling and Beyond in Computational Fluid Dynamics (CFD)
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 "". 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 "",
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 "",
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 "" algorithm extracts the advantages of the method of line and
one-stage L-W method. As a core part, the pair "" 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
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
Accelerated Piston Problem and High Order Moving Boundary Tracking Method for Compressible Fluid Flows
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 incorporated into the design of a moving boundary
tracking (MBT) method. Technically, the resolution of the accelerated piston
problem boils down to a one-sided generalized Riemann problem (GRP) solver,
which is taken as the building block to construct schemes with the high order
accuracy both in space and time. In this paper we take this into account,
together with the cell-merging approach, to propose a new family of high order
accurate moving boundary tracking methods and verify its performance through
one- and two-dimensional test problems, along with accuracy analysis
An energy-splitting high order numerical method for multi-material flows
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, we propose a new
non-oscillatory {\em energy-splitting} Godunov-type scheme for computing
multi-fluid flows in the Eulerian framework. A novel reduced version of the BN
model is introduced as the basis for the energy-splitting scheme. In comparison
with existing two-material compressible flow models obtained by reducing the BN
model in the literature, it is shown that our new reduced model can simulate
the kinetic energy exchange around material interfaces very effectively. Then a
second-order accurate extension of the energy-splitting Godunov-type scheme is
made using the generalized Riemann problem (GRP) solver. Numerical experiments
are carried out for the shock-interface interaction, shock-bubble interaction
and the Richtmyer-Meshkov instability problems, which demonstrate the excellent
performance of this type of schemes
An Efficient, Second Order Accurate, Universal Generalized Riemann Problem Solver Based on the HLLI Riemann Solver
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
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