On the first order local stability scheme for the numerical solution of time-dependent compressible flows

The diffusion, stability and monotonicity properties of the first order local stability scheme are investigated by resorting to both analysis and numerical experimentation. The scheme, deducible from either the first order Rusanov scheme or the first order van Leer scheme, is optimally stable. The magnitude of the truncation viscosity of the scheme is in between that of the Rusanov scheme and of the second order Lax-Wendroff scheme. The scheme does not, in general, yield monotonic profiles of shocks. A modified version of the local stability scheme (which preserves the local stability character at each point in the flow region and switches over to the van Leer scheme near shocks only) is proposed for practical application. Tests on one-dimensional problems show that the scheme achieves a better resolution of the flow features than is possible with the Rusanov scheme. The scheme is found to be capable of yielding a resolution which is almost equal to that of the second order schemes. Furthermore, it offers other advantages in that it resists nonlinear instabilities, is easy to programme and requires only a modest amount of storage and time on the computer.