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A parallel, high-order direct Discontinuous Galerkin (DDG) method has been developed for solving the three dimensional compressible Navier-Stokes equations on 3D hybrid grids. The most distinguishing and attractive feature of DDG method lies in its simplicity in formulation and efficiency in computational cost. The formulation of the DDG discretization for 3D Navier-Stokes equations is detailed studied and the definition of characteristic length is also carefully examined and evaluated based on 3D hybrid grids. Accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions. The capability in handling curved boundary geometry is also demonstrated. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results obtained indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable results as the widely used BR2 scheme, clearly demonstrating that the DDG method provides an attractive alternative for solving the 3D compressible Navier-Stokes equations.

With the use of temporal derivative of flux function, a two-stage temporal discretization has been recently proposed in the design of fourth-order schemes based on the generalized Riemann problem (GRP) [21] and gas-kinetic scheme (GKS) [28]. In this paper, the fourth-order gas-kinetic scheme will be extended to solve the compressible multicomponent flow equations, where the two-stage temporal discretization and fifth-order WENO reconstruction will be used in the construction of the scheme. Based on the simplified two-species BGK model [41], the coupled Euler equations for individual species will be solved. Each component has its individual gas distribution function and the equilibrium states for each component are coupled by the physical requirements of total momentum and energy conservation in particle collisions. The second-order flux function is used to achieve the fourth-order temporal accuracy, and the robustness is as good as the second-order schemes. At the same time, the source terms, such as the gravitational force and the chemical reaction, will be explicitly included in the two-stage temporal discretization through their temporal derivatives. Many numerical tests from the shock-bubble interaction to ZND detonative waves are presented to validate the current approach.

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In this paper we consider the global and local stability and instability of solutions of a scalar nonlinear differential equation with non-negative solutions. The differential equation is a perturbed version of a globally stable autonomous equation with unique zero equilibrium where the perturbation is additive and independent of the state. It is assumed that the restoring force is asymptotically negligible as the solution becomes large, and that the perturbation tends to zero as time becomes indefinitely large. It is shown that solutions are always locally stable, and that solutions either tend to zero or to infinity as time tends to infinity. In the case when the perturbation is integrable, the zero solution is globally asymptotically stable. If the perturbation is non-integrable, and tends to zero faster than a critical rate which depends on the strength of the restoring force, then solutions are globally stable. However, if the perturbation tends to zero more slowly than this critical rate, and the initial condition is sufficiently large, the solution tends to infinity. Moreover, for every initial condition, there exists a perturbation which tends to zero more slowly than the critical rate, for which the solution once again escapes to infinity. Some extensions to general scalar equations as well as to finitedimensional systems are also presented, as well as global convergence results using Liapunov techniques.1991 Mathematics Subject Classification. Primary: 34D05, 34D23, 34D20, 34C11.

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