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Multigrid and Runge-Kutta time-marching for the solution of steady adjoint equations
by
M. Giles
Oxford
In CFD analysis, it is common to use multigrid together with Runge-Kutta time-marching with local timesteps and/or local preconditioning to solve the steady flow equations. It is then natural to want to use a similar iterative solution procedure to solve the steady adjoint flow equations for the purposes of optimal design and error analysis. In formulating the adjoint equations, if one follows the discrete adjoint approach in which the adjoint operator is the transpose of the corresponding linear operator, then automatic differentiation software can be used to create the subroutines which evaluate the spatial adjoint operator. However, it is not so clear how the time-marching evolution towards the solution of these equations should be handled if one wants a procedure which is truly adjoint, in the sense that 1000 iterations of the adjoint solver will give a linear functional which is equal to that given by 1000 iterations of the linearised direct solver.
In this paper, we investigate what is required for a proper adjoint treatment of Runge-Kutta time-marching to obtain a steady-state solution. The theory is first derived for equations which are continuous in time, and then for a class of methods which includes Runge-Kutta time-marching. The analysis allows for the possibility of partial updating of some terms, as commonly used by Jameson and others in the CFD community who do not fully update the viscous fluxes at each stage. The analysis also includes the possibility of matrix preconditioning, as in Jacobi or low Mach number preconditioning, and is extended to include multigrid as well. The results of the analysis, supported by numerical experiemnts, are that one can formulate the discrete equations to obtain the same asymptotic rate of iterative convergence for the adjoint equations as for the original nonlinear equations, but the procedure does not correspond to what would normally be produced by automatic differentiation software.
http://homepages.feis.herts.ac.uk/~comqun/AD2000/Ext_Abstracts/giles.ps
Date received: March 24, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caev-07.