Manufactured geometry: solving partial differential equations for surface flow

Artificial forces are matched to solutions in the method of manufactured solutions. Yet, dominant source terms might then mask barely tested nonlinear terms in verifications. Furthermore, many geophysical models do frequently not provide for manufactured forces. A known alternative approach is to find domain shapes that render fluid dynamic systems tractable. Manufactured geometry has, though, not yet been developed into a generalized system as has been the method of manufactured solutions. Manufacturing geometry has for this work been systematically generalized and formulated. That is, tractability for arbitrary hypersurface shapes, via the addition of spatial dimensions, is enabled without having to deduce shapes apriori. As a result, manufactured bathymetries, or topographies render momentum transport equations tractable without any additional forces. Verifications of numerical solvers with analytical solutions, thus, become compatible with existing geophysical models. Standing wave solutions provide stable bathymetries whereas dynamic solutions yield dynamic bathymetries. The method is demonstrated for the incompressible Euler equation, the Euler Equation without advection, the viscous Shallow Water Equation-configuration of the Navier-Stokes Equations, and the Shallow Water Equations without advection. If the formulation of the resulting domain shape is not compact, tractability is retained by numerically integrating the domain geometry. A use case to verify numerical schemes is demonstrated, comparing the numerical algorithm of a new ocean model with analytical solutions. In principle, the method is applicable to arbitrary unstructured meshes by introducing the spatial dimension $n+1$.