Convergent filtered scheme for the Monge-Ampère Equation
This work provides a more accurate numerical method for solving the Monge-Ampère equation, a challenging nonlinear PDE, without losing theoretical convergence guarantees.
The authors propose a filtered scheme for the Monge-Ampère equation that combines a monotone convergent method with a non-monotone modification to improve accuracy while preserving convergence to the viscosity solution.
We propose an extension to our monotone and convergent method for the Monge-Ampère equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate non-monotone modification, using an appropriately chosen filter. This results in a remarkable improvement of accuracy, but without sacrificing the convergence to the unique viscosity solution.