Development of iterative techniques for the solution of unsteady compressible viscous flows
Abstract
Efficient iterative solution methods are being developed for the numerical solution of two and threedimensional compressible NavierStokes equations. Iterative time marching methods have several advantages over classical multistep explicit time marching schemes, and noniterative implicit time marching schemes. Iterative schemes have better stability characteristics than noniterative explicit and implicit schemes. Thus, the extra work required by iterative schemes can also be designed to perform efficiently on current and future generation scalable, missively parallel machines. An obvious candidate for iteratively solving the system of coupled nonlinear algebraic equations arising in CFD applications is the Newton method. Newton's method was implemented in existing finite difference and finite volume methods. Depending on the complexity of the problem, the number of Newton iterations needed per step to solve the discretized system of equations can, however, vary dramatically from a few to several hundred. Another popular approach based on the classical conjugate gradient method, known as the GMRES (Generalized Minimum Residual) algorithm is investigated. The GMRES algorithm was used in the past by a number of researchers for solving steady viscous and inviscid flow problems with considerable success. Here, the suitability of this algorithm is investigated for solving the system of nonlinear equations that arise in unsteady NavierStokes solvers at each time step. Unlike the Newton method which attempts to drive the error in the solution at each and every node down to zero, the GMRES algorithm only seeks to minimize the L2 norm of the error. In the GMRES algorithm the changes in the flow properties from one time step to the next are assumed to be the sum of a set of orthogonal vectors. By choosing the number of vectors to a reasonably small value N (between 5 and 20) the work required for advancing the solution from one time step to the next may be kept to (N+1) times that of a noniterative scheme. Many of the operations required by the GMRES algorithm such as matrixvector multiplies, matrix additions and subtractions can all be vectorized and parallelized efficiently.
 Publication:

NASA STI/Recon Technical Report N
 Pub Date:
 August 1991
 Bibcode:
 1991STIN...9127503S
 Keywords:

 Algorithms;
 Flow Characteristics;
 Iterative Solution;
 NavierStokes Equation;
 Steady Flow;
 Time Marching;
 Unsteady Flow;
 Viscous Flow;
 Conjugate Gradient Method;
 Finite Difference Theory;
 Inviscid Flow;
 Iteration;
 Matrices (Mathematics);
 Newton Methods;
 Nonlinear Equations;
 Vectors (Mathematics);
 Fluid Mechanics and Heat Transfer