The method of manufactured solutions yields exact solutions of partial differential equations by introducing artificial forcing terms. Yet many fluid dynamical models, particularly in geophysical applications, do not admit such modifications, which may also compromise physical fidelity.
In the new method of manufactured geometry this paradigm is inverted: instead of modifying the governing equations, domain geometries are constructed such that prescribed solutions satisfy the original equations exactly. In this way, exact solutions of nonlinear conservation laws are obtained without the introduction of artificial forces.
Tractability is retained even for complex hypersurface domains, while compatibility with existing physical models is preserved. The method is demonstrated for the incompressible Euler equations and viscous shallow water systems.
The proposed framework provides a new approach to verification of numerical solvers for nonlinear partial differential equations, by comparison with the constructed exact solutions.