A method for enhancing the stability and robustness of explicit schemes in astrophysical fluid dynamics

A method for enhancing the stability and robustness of explicit schemes in computational fluid dynamics is presented. The method is based in reformulating explicit schemes in matrix form, which cane modified gradually into semi or strongly-implicit schemes. From the point of view of matrix-algebra, explicit numerical methods are special cases in which the global matrix of coefficients is reduced to the identity matrix $I$. This extreme simplification leads to severer stability range, hence of their robustness. In this paper it is shown that a condition, which is similar to the Courant-Friedrich-Levy (CFL) condition can be obtained from the stability requirement of inversion of the coefficient matrix. This condition is shown to be relax-able, and that a class of methods that range from explicit to strongly implicit methods can be constructed, whose degree of implicitness depends on the number of coefficients used in constructing the corresponding coefficient-matrices. Special attention is given to a simple and tractable semi-explicit method, which is obtained by modifying the coefficient matrix from the identity matrix $I$ into a diagonal-matrix $D$. This method is shown to be stable, robust and it can be applied to search for stationary solutions using large CFL-numbers, though it converges slower than its implicit counterpart. Moreover, the method can be applied to follow the evolution of strongly time-dependent flows, though it is not as efficient as normal explicit methods. In addition, we find that the residual smoothing method accelerates convergene toward steady state solutions considerably and improves the efficiency of the solution procedure.Comment: 33 pages, 15 figure

A comparison of the LR and QR transformations for finding the eigenvalues for real nonsymmetric matrices

The LR and QR algorithms, two of the best available iterative methods for finding the eigenvalues of a nonsymmetric matrix associated with a system of linear homogeneous equations, are studied. These algorithms are discussed as they apply to the determination of the eigenvalues of real nonsymmetric matrices. A comparison of the speed and accuracy of these transformations is made. A detailed discussion of the criterion for convergence and the numerical difficulties which may occur in the computation of multiple and complex conjugate eigenvalues are included. The results of this study indicate that the QR algorithm is the more successful method for finding the eigenvalues of a real nonsymmetric matrix --Abstract, page ii

Fast methods to numerically integrate the Reynolds equation for gas fluid films

The alternating direction implicit (ADI) method is adopted, modified, and applied to the Reynolds equation for thin, gas fluid films. An efficient code is developed to predict both the steady-state and dynamic performance of an aerodynamic journal bearing. An alternative approach is shown for hybrid journal gas bearings by using Liebmann's iterative solution (LIS) for elliptic partial differential equations. The results are compared with known design criteria from experimental data. The developed methods show good accuracy and very short computer running time in comparison with methods based on an inverting of a matrix. The computer codes need a small amount of memory and can be run on either personal computers or on mainframe systems

Global model of linearized atmospheric perturbations, A: model description

Includes bibliographical references