One of the few multidimensional approaches in CFD. Combining with the entropy control concept which is even more physical than before was my PhD thesis under supervision of Dr. Farzad Ismail in USM (Universiti Sains Malaysia).
Residual-distribution (RD) methods have fundamental benefits over finite volume or finite difference methods particularly in mimicking multi-dimensional physics, achieving higher order accuracy with much smaller stencils and less sensitivity to grid changes. Thus, there is a very strong appeal in developing new numerical methods based on RD approach. The notion of discretely enforcing the Second Law of Thermodynamics (or entropy) has been around for quite some time in the CFD community although most numerical method developments vaguely achieve this important thermodynamic principle. For instance, most CFD methods or codes are deemed to be entropy-satisfying if they minimally produce entropy for smooth flows and generates entropy with the correct sign across a shock.
Prior to 2006, almost all of the work in entropy-control are chiefly academic and not practical numerically. They are either numerically expensive or not well-posed. However, in 2006 Ismail and Roe developed an explicit, well-posed and relatively cheap entropy-conserved and entropy-stable finite volume method for the system of Euler equations based on a one dimensional residual-distribution approach. Since then, the topic of numerical entropy-control has been developed into Magnetohydrodynamics, Shallow Water Equations, Incompressible Navier-Stokes and Compressible Navier-Stokes equations using the original idea of Ismail and Roe. Unfortunately, almost all studies of entropy-control are based on finite volume methods.
For the scalar part, by controlling the entropy we could get first and second order accurate in linear and nonlinear cases. Also, for the Euler equations, we also generate low and high order results.
The entropy generation is controlled by parameter (q). Subsonic flow results are shown here. For the q=0.7 which is showing the low order solution compared with the classic N, we see some improvement in the back of the cylider in terms of less diffusion. For q=1.5 demonstrating the second order the result is compared with the classic LDA with compact stencil.