The construction of efficient high order low dissipation numerical methods for nonlinear conservation laws has been the subject of much research recently. For smooth flows, it is well known that the standard high order non-dissipative central schemes generate spurious noise leading to nonlinear instability, especially for long time integration applications such as in aeroacoustics, rotorcraft dynamics, and turbulence physics. On the other hand, central schemes in conjunction with linear numerical dissipations are too diffusive for the physical problems in question. At the same time the majority of the available high order high-resolution shock-capturing schemes are too CPU intensive for practical computations. In spite of their high-resolution capability for rapidly evolving flows and short term time integrations, for long time integrations these schemes often exhibit undesirable amplitude errors for aeroacoustics and turbulence computations. Current focus has been mainly on algorithmic issues in constructing highly accurate methods away from boundaries. Rigorous stability estimates for accurate and appropriate numerical boundary conditions associated with fourth-or higher-order methods are equally important and have been the major stumbling block in the theoretical development of these schemes for nonlinear systems. Most of the existing theory for nonlinear conservation laws and their finite discretizations is concerned with the initial value problem (IVP). Standard practice in computational fluid dynamics (CFD) involving boundary conditions relies on guidelines from theory for linear stability analysis of initial boundary value problems (IBVPs) or IVP theories with the boundary conditions ignored. These linearized stability guidelines are only necessary but not sufficient for nonlinear stability. Spatial nonlinear stability of IBVPs goes hand-in-hand with the appropriate amount of nonlinear numerical dissipation required to stabilize the spatial scheme. The delicate balance of the numerical dissipation for stability without the expense of excessive smearing of the flow physics after long time integrations poses a severe challenge for unsteady flow simulations of this type. Actually, there are two possible sources of stabilizing mechanisms, namely, (a) from the governing equation level and (b) from the numerical scheme level. Employing a nonlinear stable form of the governing equations (more conditioned form of the PDE) in conjunction with the appropriate nonlinear stable scheme for IBVPs is one way of minimizing the use of numerical dissipation.
Until recently it was not known how to derive the proper numerical boundary conditions for a rigorous stability estimate for conventional spatial high order central difference schemes for nonlinear hyperbolic IBVPs. Advances by Kreiss and Scherer [1], Strand [2], and Olsson [3] led to the construction of high order boundary difference operators that enabled the design of stable high order central schemes for linear hyperbolic systems. The major tool used to overcome the stumbling block is a generalized energy method. The basic building block in establishing a stable energy estimate for high order spatial central