A Method to Calculate Numerical Errors Using Adjoint Error Estimation for Linear Advection

This paper is concerned with the computation of numerical discretization error for uncertainty quantification. An a posteriori error formula is described for a functional measurement of the solution to a scalar advection equation that is estimated by finite volume approximations. An exact error formula and computable error estimate are derived based on an abstractly defined approximation of the adjoint solution. The adjoint problem is divorced from the finite volume method used to approximate the forward solution variables and may be approximated using a low-order finite volume method. The accuracy of the computable error estimate provably satisfies an a priori error bound for sufficiently smooth solutions of the forward and adjoint problems. Computational examples are provided that show support of the theory for smooth solutions. The application to problems with discontinuities is also investigated computationally.