Partial Differential Equations (PDEs) play a ubiquitous role in scientific computing and engineering. While numerical methods make solving PDEs tractable, these numerical solvers encounter several issues, particularly for hyperbolic PDEs. These issues arise from multiple sources including the PDE’s physical model, which can lead to effects like shock wave formation, and the PDE solver’s inherent approximations, which can introduce spurious numerical artifacts. These issues can cause the solver’s program execution to crash (due to overflow) or return results with unacceptable levels of inaccuracy (due to spurious oscillations or dissipation). Moreover, these challenges are compounded by the nonlinear nature of many of these PDEs. In addition, PDE solvers must obey numerical invariants like the CFL condition. Hence there exists a critical need to apply program analysis to PDE solvers to certify such problems do not arise and that invariants are always satisfied. As a solution, we develop Phocus, which is the first abstract interpretation of hyperbolic PDE solvers. Phocus can certify precise bounds on nonlinear PDE solutions and certify key invariants such as the CFL condition and a solution’s total variation bound. Hence Phocus can verify the absence of shock formation, the stability of the solver, and bounds on the amount of spurious numerical effects. To enable effective abstract interpretation of hyperbolic PDE solvers, Phocus uses a novel optimization-based procedure to synthesize precise abstract transformers for multiple finite difference schemes. To evaluate Phocus, we develop a new set of PDE benchmark programs and use them to perform an extensive experimental evaluation which demonstrates Phocus’s significant precision benefits and scalability to several thousand mesh points.

@article{paperbot2912,
  title = {Synthesizing Sound and Precise Abstract Transformers for Nonlinear Hyperbolic PDE Solvers},
  author = {Jacob Laurel and Ignacio Laguna and Jan Hückelheim},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {9},
  number = {OOPSLA2},
  year = {2025},
  doi = {10.1145/3763088}
}