Symmetry-preserving upwind discretization of convection on non-uniform grids

Abstract

Although upwind discretization of convection will lead to a diagonally dominant coefficient matrix, on arbitrary grids the latter is not necessarily positive real, i.e. its symmetric part need not be positive definite (‘negative diffusion’). Especially on contracting-expanding grids this property can be lost. The paper discusses a conservative (finite-volume) upwind variant for which the latter property is guaranteed to hold, irrespective of grid (ir)regularity. Further, empirically it is found that often its global discretization error is smaller than that of the ‘traditional’ (finite-difference) upwind method. Finally, it is shown that in many situations its extremal eigenvalues at the outer side of the spectrum move towards the imaginary axis, thus enhancing the stability of explicit time-integration methods.