First, second, and third order finite-volume schemes for advection–diffusion

In this paper, we present first, second, and third order implicit finite-volume solvers for advection–diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate third-order advection–diffusion scheme is made trivial by the hyperbolic method while a naive construction of adding a third-order diffusion scheme to a third-order advection scheme can fail to yield third-order accuracy. We demonstrate also that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable on irregular triangular grids: first, second and third order accurate gradients by the first, second, and third order schemes, respectively. Furthermore, the first and second order schemes are shown to achieve one order higher accuracy for the solution variable in the advection limit. It is also shown that these schemes are capable of producing highly accurate and smooth solution gradients along the boundary in a highly-skewed anisotropic irregular triangular grid while conventional schemes suffer from oscillations on such a grid. Numerical results show that these schemes are capable of delivering high accuracy over conventional schemes at a significantly reduced cost.