New Scheme for the Computation of Compressible Flows

A new flux-based scheme for compressible flows

“…The present work attempts to extend the previously reported PVU schemes [23,24] by incorporating higher order central / upwind biased interpolations for both inter-cell velocity and convection transport property vector. The various higher order and lower order interpolations are then combined in a convex manner through newly designed normalized weight functions that control the weight of lower / higher order interpolations on the basis of scaled convection velocity magnitude and the spatial rate of change of solution gradient or second spatial derivative of the solution.…”

“…The performance of the second order scheme, as represented by accuracy, global convergence rates, CPU time versus accuracy statistics was not examined in detail through test cases involving standard scalar hyperbolic conservation laws in 1D (LAE, IBE) and vector hyperbolic conservation laws as represented by Euler equations in 1D as well as in multi-dimensional scenarios spanning a wide range of Mach numbers. More importantly, to place the second order PVU scheme in correct perspective, the performance of the second order scheme vis-a-vis some of the well known higher order schemes of both FVS type and Reconstruction-evolution type has not been examined in detail in the earlier works [23,24].…”

“…This is in contrast to the more conventional approach of carrying out upwinding of the split flux components on the basis of characteristic speeds for compressible flows. It was shown by Qamar et al [23] that employing first order upwind interpolation for convective transport vector eliminated the need of any kind of fluxlimiter to resolve the shocks and the flow field in their neighbourhood accurately. The stability and TVD properties were also reported.…”

“…A step in this direction was taken by Qamar et al [23] by proposing a new two-step predictor corrector scheme in which the flux vectors were split into convective and non-convective parts. The derivatives of these flux vectors were discretized in space using explicit central / forward / backward differencing stencils (overall second order spatial discretizations).…”

“…The weight function sensitive to second spatial derivative of the solution also helps to delineate the regions of smoothly varying solution from those containing discontinuities. The essential idea is to retain some portion of the higher order interpolation estimate in the numerical inter-cell flux to reduce the large dissipation associated with first order upwind interpolation estimate employed in earlier versions reported in [23,24]. The performance of the new modified PVU family of the scheme, herein after referred to as PVU-M+, is examined in detail using both scalar (LAE and IBE) and vector hyperbolic conservation laws in 1D as well as for multi-dimensional, inviscid / viscous, nearly incompressible to very high speed supersonic flow test cases.…”