Goal-oriented a posteriori estimation of numerical errors in the solution of multiphysics systems

This paper develops a general methodology for a posteriori error estimation in time-dependent multiphysics numerical simulations. The methodology builds upon the generalized-structure additive Runge–Kutta (GARK) approach to time integration. GARK provides a unified formulation of multimethods that simulate complex systems by applying different discretization formulas and/or different time steps to individual components of the system. We derive discrete GARK adjoints and analyze their time accuracy. Based on the adjoint method, we establish computable a posteriori identities for the impacts of both temporal and spatial discretization errors on a given goal function. Numerical examples with reaction-diffusion systems illustrate the accuracy of the derived error measures. Local error decompositions are used to illustrate the power of this framework in adaptive refinements of both temporal and spatial meshes.