Adjoint-guided Adaptive Mesh Refinement for Hyperbolic Systems of Equations

Book excerpt: One difficulty in developing numerical methods for time-dependent partial differential equations is the fact that solutions contain time-varying regions where much higher resolution is required than elsewhere in the domain. The open source Clawpack software implements block-structured adaptive mesh refinement to selectively refine around propagating waves in the AMRClaw and GeoClaw packages. In particular, GeoClaw is widely used for tsunami modeling, the application that motivated this work. For problems where the solution must be computed over a large domain but is only of interest in one small area (e.g. one coastal community when doing tsunami modeling, or the location of a pressure gauge when doing acoustics modeling), a method that allows identifying and refining the grid only in regions that influence this target area would significantly reduce the computational cost of finding a solution. The adaptive mesh refinement approach currently implemented in AMRClaw and GeoClaw often refines waves that will not impact the target area. To remedy this, we seek a method that enables the identification and refinement of only the waves that will influence the location of interest. In this work we show that solving the time-dependent adjoint equation and using a suitable inner product with either the forward solution, or the estimated one-step error in the forward solution, allows for a more precise refinement of the relevant waves. We present the adjoint methodology first in one space dimension for illustration and in a broad context since it could also be used in other adaptive software, and for other tsunami applications beyond adaptive mesh refinement. We then show how this adjoint method has been integrated into the adaptive mesh refinement strategy of the open source AMRClaw and GeoClaw software and present linear variable coefficient acoustics and tsunami modeling results showing that the accuracy of the solution is maintained and the computational time required is significantly reduced through the integration of the adjoint method into adaptive mesh refinement. The adjoint method is compared to adaptive mesh refinement methods already available in the AMRClaw software, and the advantages and disadvantages of using the adjoint method are discussed. Other capabilities of the adjoint method such as focusing on specific time ranges of interest, sensitivity analysis, and source impact analysis and design are also presented. The new algorithms are incorporated in Clawpack and code for the examples presented in this work is archived on Github.