A natural derivation of discontinuity capturing operator for convection–diffusion problems Original Research Article

The concept of effective transport velocity is introduced to derive a discontinuity capturing operator for convection–diffusion problems. The effective transport velocity, which depends both on the flow velocity and on the local solution gradient, is used to modify the classical representation of the convective term at the continuum level. As a result, a discontinuity capturing operator arises naturally in the derivation of Lax–Wendroff, Taylor–Galerkin and least-squares type approximations of the convection–diffusion equation. The numerical examples presented demonstrate the effectiveness of the proposed approach. These include the classical problem of the advection of a steep profile skew to the mesh and the computation of the temperature field in a free convection problem.