Comparing Matrix-based and Matrix-free Discrete Adjoint Approaches to the Euler Equations

Detail is presented on the implementation of numerical derivatives with focus given to the discrete adjoint equations. Two approaches are considered: a hybrid matrix-based scheme where the convective Jacobian is constructed explicitly; and a matrix-free method using reverse-mode automatic differentiation. The hybrid matrix-based scheme exploits a compact convective stencil using graph colouring to evaluate the convective Jacobian terms in $O(10)$ residual evaluations. Jacobian terms, grouped by colours, are evaluated using the complex step tangent model; this approach requires no external libraries or tools, minimal code modification and provides derivatives accurate to machine precision. The remaining artificial dissipation terms are trivial to differentiate by hand where the sensor coefficients are held constant. The hybrid matrix-based methodology is validated and compared with the `traditional` matrix-free approach using reverse-mode automatic differentiation. The adjoint equations using both approaches are solved using the same fixed-point Runge-Kutta iteration accelerated by agglomeration multigrid. No loss in accuracy is seen between the matrix-based and the matrix-free methods when validated with the complex step tangent model. The hybrid matrix-based approach demonstrates a notable runtime performance advantage over the traditional matrix-free approach due to the prior calculation of Jacobian terms. Moreover, the convective Jacobian calculation takes less than $5\%$ of primal runtime due to the compact stencil used. A critical analysis of the results and methodology is consequently presented, focusing on the general applicability of the hybrid approach to more complex problems.