An exact discretization for a transport equation with piecewise-constant coefficients and arbitrary source

The search for a robust and accurate discretization scheme for a general convection diffusion equation has been a driving force in the improvement of codes employed in the CFD community. As part of this avenue of inquiry the research in this paper was aimed at improving the accuracy of schemes implemented in an in-house RANS code by incorporating the source in the discretization stencil. To understand how to do this we started working with the equation for a transported scalar and we came up with a systematic way of transforming a source-dominant equation into a source-free equation that allowed us to obtain an exact solution, i.e., a solution whose error is within machine accuracy, for an arbitrary source and an arbitrary number of discretization nodes, as long as the coefficients are piecewise constant in the intervals. The final equation, being a source-free equation, allows the use of the exponential scheme for the coefficients which is known to be exact for the homogeneous 1D convection diffusion equation. This paper shows how to convert the exponential scheme into an exact scheme for a transport equation with piecewise constant coefficients and arbitrary source in a 1D domain.

► We developed a new discretization for convection diffusion equations with arbitrary sources.
► The exact results are independent of the number of nodes.
► The mass flux and diffusivity coeffcients have to be constant and the source continuous in the interval.
► Results reach machine accuracy with just one node.