2-D/axisymmetric formulation of multi-dimensional upwind scheme

A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes

“…But on the other hand, its application to diffusion problems had long been almost untouched, apparently because diffusion is an isotropic process and does not benefit particularly from such a multidimensional capability. In fact, it has been a standard practice to discretize the viscous term by the Galerkin method and simply add to the existing residual-distribution Euler code to construct a Navier-Stokes code [15][16][17]. It was pointed out in [18] however that such a strategy deteriorated the formal accuracy of the scheme due to an incompatibility of the two discretizations, especially in regions where advection and diffusion effects are equally important.…”

Finite element/volume solution to axisymmetric conservation laws

“…Remarkably enough, the expression of the mass-lumped approximation (12) is very similar to its finite volume counterpart (15); in fact, they are formally equivalent -provided that the finite element metric quantities L y i , g y ik and n y i can be linked to their finite volume counterparts V y i , m y ik and m y i -but for the presence in (12) of the domain termŷ Á ðf k À f i ÞM ik =2 and the boundary term ðf k À f i Þ Á v y ik =2, which are regarded here as higherorder corrections to (15) stemming form the piecewise linear finite element representation of the unknown. Note that the higher-order corrections vanish if these contributions are diagonalized or lumped.…”

“…This is a key feature of the scheme that allows to automatically account for the axis of symmetry without the need of including numerical switch or special treatments. This is not the case for example in the FV schemes proposed in [15,17], in which the condition of a zero normal flux at the axis of symmetry is imposed explicitly.…”

“…The scheme (15) is now compared to the finite element approximation of the scalar conservation law (3).…”

“…This form of the equation is the key for building the solver for axially symmetric problems straightforwardly upon an existing 2D Cartesian solver for plane problems. The same form has been considered by Wood and Kleb in the development of a FV fluctuation splitting method for axisymmetric compressible flows [15]. It is also worth recalling that a different approach, termed the domain perturbation, has been followed by Goudjo and Desideri in which the scheme is derived as the limiting case of a three-dimensional FV discretization for a control volume of vanishing thickness [17].…”