An automated approach for developing discrete adjoint solvers

An automatic differentiation tool, Tapenade, is used to develop the adjoint node for a three-dimensional computational fluid dynamics (CFD) solver. Rather than using automatic differentiation to differentiate the complete source code of the CFD solver (including the iterative loop), we have applied it selectively to produce code that computes both the flux Jacobian matrix that is present iti the discrete adjoint matrix and the terms in the gradient computation that ore necessary for this type of formulation. The resulting linear discrete adjoint system is then solved using the toolkit for parallel scientific computations, PETSc. This approach to deriving and implementing the discrete adjoint equations has the advantages that it is applicable to arbitrary sets of governing equations and cost functions, it is exactly consistent with the gradients that would be computed by an infinitely-accurate finite difference of the original solver, and it is automatic, thus avoiding the lengthy development times required to develop discrete adjoint solvers for complex governing equations. All of these significant advantages corne at the cost of increased memory requirements for the discrete adjoint solver. However, given typical use patterns for today's parallel computers, this disadvantage is rather small when compared with the very significant advantages that are obtained. The approach is outlined in this paper together with some preliminary results of the implementation of these ideas.