An accurate discretization for an inhomogeneous transport equation with arbitrary coefficients and source

• A three-node algebraic equation for arbitrary coefficients and source of second-order ODE has been derived.
• It is exact.
• The coefficients involve integrals that can be calculated with high accuracy via Hermite splines.
• The results show order of convergence very high and errors for high Péclet much less than with other traditional schemes.

A new way of obtaining the algebraic relation between the nodal values in a general one-dimensional transport equation is presented. The equation can contain an arbitrary source and both the convective flux and the diffusion coefficient may vary arbitrarily. Contrary to the usual approach of approximating the derivatives involved, the algebraic relation is based on the exact solution written in integral terms. The required integrals can be speedily evaluated by approximating the integrand with Hermite splines or applying Gauss quadrature rules. The startling point about the whole procedure is that a very high accuracy can be obtained with few nodes, and more surprisingly, it can be increased almost up to machine accuracy by augmenting the number of quadrature points or the Hermite spline degree with little extra cost.