Local preconditioners for steady and unsteady flow applications

Abstract

Abstract. Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners: Jacobi and low speed preconditioning. We can express the algorithm in several sets of variables while using only the conservation variables for the flux terms. We compare the effect of these various variable sets on the efficiency and accuracy of the scheme.Mathematics Subject Classification. 65M06, 76M12.

An all-speed relaxation scheme for gases and compressible materials

“…The relaxation matrix A is diagonal and it is built by imposing the subcharacteristic condition (20). The wave speeds of the relaxation system are the following…”

“…Condition (20) states that the eigenvalues λ i of the original system need to lie between the eigenvalues µ j of the relaxation system.On the other hand, the CFL constraint has to be enforced on the speeds (21) of the relaxation system. Therefore, the smallest A satisfying (20) is needed.…”

“…A possible approach consists in choosing implicit time discretizations, which allow to avoid the acoustic CFL constraint. Several preconditioning techniques have been devised to decrease the numerical diffusion of upwind schemes at low Mach numbers in such implicit methods [14,15,16,17], starting from the "artificial compressibility" technique of Chorin [18] and from Turkel's work [19,20]. However, it is extremely difficult to handle the non-linearities of classical upwind discretizations (e.g.…”

Estimation of goal functional error arising from iterative solution of Euler equations

“…This approach implies iterations along the time coordinate or a certain pseudo-temporal variable [4][5]. Different variants of preconditioning [6][7][8][9][10] are used to improve the relaxation rate. Commonly used a priori estimates of iteration convergence [5] (linking the error with iteration residual in certain norms) contain constants that are in general case (for nonlinear non selfadjoint operators) unknown.…”

Computational-Fluid-Dynamics Solver with Preconditioned Method for Aerodynamic Simulation of High-Lift Configuration

“…One typical solution is the use of Riemann BCs, which are based on the one-dimensional characteristics to eliminate the spurious reflecting waves. A characteristic form of the preconditioned system of equations has been given [11,14,15], and preconditioned characteristic BCs are proposed for the preconditioned methods proposed by Ref. [12].…”