Implications of Using Dual Number Derivatives With a Numerical Solution

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.