An algorithm for ideal multigrid convergence for the steady Euler equations

On the combined performance of nonlocal artificial boundary conditions with the new generation of advanced multigrid flow solvers

“…The main advantage of this approach is its simplicity, it can also be classified as Weighted Gauss-Seidel relaxation [10]. Subsequent work in this direction is presented in [15,16]. The limitation of this approach, however, is that it is not clear as of yet whether or not it can be generalized to the case of viscous compressible flows.…”

“…It, however, turns out, see [17], that the idea of factorization that has successfully led to the construction of essentially optimal multigrid solvers in [10][11][12][13][14][15][16] cannot be applied to an important class of schemes that is routinely used for compressible flow computations -the shock-capturing schemes of all types, including the original formulation of genuinely multidimensional scheme, see [7][8][9]. As these schemes possess a collection of features that cannot be either compromised or traded for in computational practice, primarily the mechanism for handling discontinuous solutions, there is a need for discretization that would rather add the capability of factoring the governing PDEs to the existing properties of the scheme.…”

“…We emphasize that "the simplest existing versions" does not always mean the latest most universal and advanced ones, but rather those that have been tested and that most easily lend themselves to the analysis by analytical means. Specifically, we employ the so-called pressure-Poisson formulation of the factorizable scheme [14][15][16] for calculating the two-dimensional inviscid incompressible fluid flow around an airfoil. Analytical solutions for such flows are readily available through the use of the conformal mapping technique.…”

“…It was realized later that deriving the discrete PPE directly can simplify greatly the treatment of the boundary conditions and also introduce some desirable features into the scheme (like a certain conservation property that appeared to be crucial for the purpose of the current work, see Section 2.2.1). This algorithm is described in detail for both unstructured triangular and structured quadrilateral grids in [15]. Computational results obtained with this algorithm have been presented previously in [14,16].…”

“…and dp 'dp iK*), (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) respectively. The external artificial boundary on the plane C, will be a circle of radius R centered at the origin: p = R. As we will see in Section 3.1, the grid used for the numerical integration of the flow equations is obtained by transforming a polar grid from the plane C back to the plane z.…”