SUPG-YZβ formulation for solving convection-dominated steady linear reaction-convection-diffusion equations

In this computational study, stabilized finite element solutions of convection-dominated steady linear reaction-convection-diffusion equations are examined. Although the standard Galerkin finite element method (GFEM) is one of the most robust, efficient, and reliable methods for many engineering simulations, it suffers from instability issues in solving convection-dominated problems. To this end, this work deals with a stabilized version of the standard GFEM, called the streamline-upwind/Petrov-Galerkin (SUPG) formulation, to overcome the instability issues in solving such problems. The stabilized formulation is further supplemented with YZβ shock-capturing to provide additional stability around sharp gradients. A comprehensive set of test computations is provided to compare the results obtained by using the GFEM, SUPG, and SUPG-YZβ formulations. It is observed that the GFEM solutions involve spurious oscillations for smaller values of the diffusion parameter, as expected. These oscillations are significantly eliminated when the SUPG formulation is employed. It is also seen that the SUPG-YZβ formulation provides better solution profiles near steep gradients, in general.

ANALYSIS OF THE SPACE SEMI-DISCRETIZED SUPG METHOD FOR TRANSIENT CONVECTION–DIFFUSION EQUATIONS

We consider space semi-discretization of the transient convection–diffusion equation using a Streamline Upwind Petrov–Galerkin (SUPG) finite element method. We show stability and convergence both for the standard SUPG formulation with elementwise evaluated Laplacian and for a weakly consistent formulation where the elementwise Laplacian is replaced by a discrete, reconstructed, Laplacian. Some numerical examples are presented to illustrate the theory.

Computation of Inviscid Supersonic Flows Around Cylinders and Spheres With the V-SGS Stabilization and YZβ Shock-Capturing

The YZβ shock-capturing technique was introduced originally for use in combination with the streamline-upwind/Petrov–Galerkin (SUPG) formulation of compressible flows in conservation variables. It is a simple residual-based shock-capturing technique. Later it was also combined with the variable subgrid scale (V-SGS) formulation of compressible flows in conservation variables and tested on standard 2D test problems. The V-SGS method is based on an approximation of the class of SGS models derived from the Hughes variational multiscale method. In this paper, we carry out numerical experiments with inviscid supersonic flows around cylinders and spheres to evaluate the performance of the YZβ shock-capturing combined with the V-SGS method. The cylinder computations are carried out at Mach numbers 3 and 8, and the sphere computations are carried out at Mach number 3. The results compare well to those obtained with the YZβ shock-capturing combined with the SUPG formulation, which were shown earlier to compare very favorably to those obtained with the well established OVERFLOW code.

Three-Dimensional Edge-Based SUPG Computation of Inviscid Compressible Flows With YZβ Shock-Capturing

The streamline-upwind/Petrov–Galerkin (SUPG) formulation of compressible flows based on conservation variables, supplemented with shock-capturing, has been successfully used over a quarter of a century. In this paper, for inviscid compressible flows, the YZβ shock-capturing parameter, which was developed recently and is based on conservation variables only, is compared with an earlier parameter derived based on the entropy variables. Our studies include comparing, in the context of these two versions of the SUPG formulation, computational efficiency of the element- and edge-based data structures in iterative computation of compressible flows. Tests include 1D, 2D, and 3D examples.