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Due to its excellent shock-capturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristic-wise construction method should be applied. Compared to component-wise reconstruction, characteristic-wise reconstruction leads to much higher computational cost and thus is not suitable for large scale simulation such as direct numerical simulation of turbulence. In this paper, an adaptive characteristic-wise reconstruction WENO-Z scheme, i.e. the AdaWENO-Z scheme, is proposed to improve the computational efficiency of the characteristic-wise reconstruction method. By defining shared smoothness functions, shared smoothness indicators as well as shared WENO weights are firstly introduced to reduce the computational cost of the component-wise reconstruction procedure and to define a global switch function capable of detecting discontinuity. According to the given switch function, the new scheme performs characteristic-wise reconstruction near discontinuities and switches to component-wise reconstruction for smooth regions. Several one dimensional and two dimensional numerical tests are performed to validate and evaluate the AdaWENO-Z scheme. Numerical results show that AdaWENO-Z maintains essentially non-oscillatory flow field near discontinuities as with the characteristic-wise reconstruction method. Besieds, compared to the component-wise reconstruction method, AdaWENO-Z is about 20% to 40% faster which indicates its excellent efficiency.

We prove the global-in-time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data ψ0,A0 ∈ L3,u0 ∈ L3 ∕ 2,u0 ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we present a high-order moving mesh (HMM) kinetic scheme for compressible flow computations on unstructured meshes. To construct the scheme, we employ the frame of the remapping-free ALE-type kinetic method (Ni et al. in J Comput Phys 228:3154---3171, 2009) to get the discretization of compressible system. For the space accuracy, we use the weighted essential non-oscillatory reconstruction on the adaptive moving mesh from Tang and Tang (SIAM J Numer Anal 41:487---515 2003) to achieve time accuracy,we make use of the kinetic flux which includes time accurate integral, and thus obtain a HMM scheme. A number of numerical examples are given, especially an isentropic vortex problem to show the convergence order of the scheme. Numerical results demonstrate the accuracy and robustness of the scheme.

SUMMARY On the basis of the work [P.-H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell-centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell-centered. And using the nodal solver, we obtain the nodal viscous-velocity, viscous-pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous-velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One- and two-dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.

We investigate the nonlinear instability of a smooth steady density profile solution of the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, this is different from the previous known results., 24 pages

Based on an efficient Riemann solver, a remapping-free ALE method (RALE) for multi-material fluids with general equations of state is proposed. The basic idea of constructing the RALE is to couple the Lagrangian method with a remapping-free ALE-type method. In order to keep the sharpness of a material interface, the Lagrangian formulation is employed for tracking the material interface, where the Lagrangian velocity of nodes and Lagrangian fluxes are designed. In single material regions, the numerical fluxes are constructed on moving meshes which move nodes to the regions with large gradients to increase the numerical accuracy, and the explicit remapping stage is avoided because of the new discrete scheme. The inverse Hermite interpolation argument is employed in solving the Riemann problem with general EOS, consequently, reducing iteration steps greatly and resulting in an efficient and robust Riemann solver. A number of numerical examples are presented.

In this paper,a moving mesh BGK scheme (MMBGK) for multi-material flow computa- tions is proposed. The basic idea of cons- tructing the MMBGK is to couple the Lagran- gian method for the material inter- face with the remapping-free ALE-type kinetic method within each single material region. Numerical examples in one and two dimensional space are presented to illustrate the efficiency and accuracy of the scheme.

The paper presents a Discontinuous Galerkin γ-BGK (γ-DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a γ-BGK scheme based on WENO limiters. In this γ-DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high-order accuracy and sharp non-oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright © 2010 John Wiley & Sons, Ltd.

This paper presents a discontinuous Galerkin BGK (DGBGK) method for both viscous and inviscid flow simulations under a DG framework with a gas-kinetic flux and WENO limiters. In the DGBGK method, the construction of the flux in the DG method is based on the particle transport and collisional mechanism which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation. Due to the connection between the gas-kinetic BGK model and the Euler as well as the Navier-Stokes equations, both viscous and inviscid flow equations can be simulated by a unified formulation. WENO limiters are used to obtain uniform high-order accuracy and sharp non-oscillatory shock transition. In the current method, the time accuracy is achieved by the direct integration of both time-dependent flux function at a cell interface and the flow variables inside each element. Numerical examples in one and two space dimensions are presented to illustrate the robustness and accuracy of the present scheme.

This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements.

We present a second-order @c-model BGK scheme for compressible multimaterial flows, which extends the authors' earlier work on a first-order scheme [S. Jiang, G. Ni, A @c-model BGK scheme for compressible multifluids, Internat. J. Numer. Methods Fluids 46 (2004) 163-182]. The scheme is based on the incorporation of a conservative @c-model scheme given in [R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: A quasi conservative approach, J. Comput. Phys. 125 (1996) 150-160] into the gas kinetic BGK scheme [K.H. Prendergast, K. Xu, Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys. 109 (1993) 53-66; K. Xu, K.H. Prendergast, Numerical Navier-Stokes solutions from gas kinetic theory, J. Comput. Phys. 114 (1994) 9-17]. Several numerical examples presented in this paper validate the scheme in numerical simulations of compressible multifluids.

We prove the solution of the Navier--Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength exists globally in time, and moreover, as the viscosity and heat-conductivity coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

We present a γ-model BGK scheme for the numerical simulation of compressible multifluids. The scheme is based on the incorporation of a conservative γ-model scheme given in (J. Comput. Phys. 1996; 125:150–160) into the gas kinetic BGK scheme (J. Comput. Phys. 1993; 109:53–66, J. Comput. Phys. 1994; 114:9–17), and is simple to implement. Several numerical examples presented in this paper validate the scheme in the application of compressible multimaterial flows. Copyright © 2004 John Wiley & Sons, Ltd.

We study the Cauchy problem for the n-dimensional Navier–Stokes equations ( n ⩾ 3 ), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.