Simulation of Parabolic Flow on an Eye-Shaped Domain with Moving Boundary
This work addresses the numerical challenge of simulating tear film dynamics on the ocular surface, which is important for ophthalmology, but the method is demonstrated only on simplified equations and is thus incremental.
The authors developed a spectral collocation method using conformal maps to simulate parabolic flow on an eye-shaped domain with a moving boundary, achieving excellent accuracy on diffusion equations as measured pointwise or by conservation checks.
During the upstroke of a normal eye blink, the upper lid moves and paints a thin tear film over the exposed corneal and conjunctival surfaces. This thin tear film may be modeled by a nonlinear fourth-order PDE derived from lubrication theory. A challenge in the numerical simulation of this model is to include both the geometry of the eye and the movement of the eyelid. A pair of orthogonal and conformal maps transform a square into an approximate representation of the exposed ocular surface of a human eye. A spectral collocation method on the square produces relatively efficient solutions on the eye-shaped domain via these maps. The method is demonstrated on linear and nonlinear second-order diffusion equations and shown to have excellent accuracy as measured pointwise or by conservation checks. Future work will use the method for thin-film equations on the same type of domain.