Analysis and design of preconditioning methods for the Euler equations.

摘要

Preconditioning of a system of equations at the differential level represents a relatively new area of research in convergence acceleration of discrete schemes for the fluid dynamics equations. This technique attempts to remove the intrinsic stiffness of the equations caused by the different time scales of the dynamic problem. Specifically, for the Euler equations, preconditioning aims at equalizing the speed of propagation of the different waves of the hyperbolic problem. This ‘fix’ becomes particularly useful in the incompressible limit and in the neighborhood of the sonic point. Practical examples of application of preconditioning include modern transonic supercritical airfoils, and nearly incompressible flows with embedded regions where compressibility is important (e.g., low speed combustion).; This study attempts the ambitious project of reviewing most of the work done in Euler preconditioning, while at the same time extending some of the existing methods and correcting their robustness problems. Several original contributions to the theory of Euler preconditioning are given, including a thorough exposition of the symmetrizability property of the preconditioned equations, a discussion of the positive definiteness property of the preconditioning matrix, and the study of the eigenvalues for the full form of the preconditioner.; Considering the numerical implementation of the preconditioned methods using the Roe flux function, a scheme based on the classical one-Riemann problem normal to the cell interface is proposed, and its advantages over other formulations found in the literature, as well as its drawbacks, are discussed. Then, the analysis of several existing preconditioning methods is conducted, and the complete eigenvector structure of the equations preconditioned with the Van Leer- Lee-Roe matrix, the Turkel matrix, the Choi-Merkle matrix, and a few others, is obtained and analyzed.; A comprehensive exploration of preconditioning in one and two spatial dimensions is attempted, which allows to better understand the properties of existing preconditioners. While this investigation suggests new interesting families of one-dimensional preconditioners, for the two-dimensional case the analysis is complete only for the sparse form of the preconditioner, and shows that in this instance it is not possible to remove all of the robustness problems usually found in Euler preconditioning. When considering the full form of the preconditioning matrix the analysis is not complete, because of the formidable algebraic problem involved. Nonetheless, some specific solutions are considered, and a few general conclusions are also drawn in this case.; Finally, it is shown that there exist at least two sparse preconditioners that are sufficiently robust in computing stagnation point flow, while preserving the overall effectiveness of preconditioning for low speed flow. One of these matrices is a modification of the popular Turkel method. Using this matrix in regions of low Mach number, in conjunction with the Van Leer-Lee-Roe preconditioner in the transonic and supersonic parts of the flow field, allows to achieve a very robust and efficient preconditioning procedure for the entire Mach regime.