An Unsteady Entropy Adjoint Approach for Adaptive Solution of the Shallow-Water Equations

This paper presents a novel approach to solution-based adaptation for unsteady discretizations of symmetrizable conservation laws. This approach is based on an extension of the entropy adjoint approach, which was previously introduced for steady-state simulations. Key to the approach is the interpretation of symmetrizing entropy variables as adjoint solutions for an output that states the entropy balance in the space-time computational domain. This relationship is shown for general rst-order conservation laws, and it is applied to the case of the shallow water equations. Speci cally, the entropy variable weighted residual is used to drive an adaptive indicator that targets regions of the space-time domain where spurious entropy generation is greatest. The error estimation and adaptation strategies are the same as those prescribed by output-based theory, with the advantage that no separate adjoint solution is required. Results for the unsteady shallow water equations in both one and two spatial dimensions show that the adaptive indicator performs better than uniform re nement and as well as or better than an unweighted residual indicator.