On the convergence of solving a nonlinear Volterra-type integral equation for surface divergence based on surface thermal information

We analyze a nonlinear integral equation for calculating free-surface divergence that was proposed by Szeri (2017, https://doi.org/10.1002/2016JC012312). When given the temperature and heat flux at a free surface, the surface divergence can be calculated through a nonlinear singular Volterra-type integral equation. The two given functions in the integral equation satisfy auxiliary conditions through a higher dimensional partial differential equation. We prove the existence and uniqueness of the solution of the integral equation. We also prove the local linear convergence of the corresponding Picard iteration method for solving the integral equation when the surface heat flux is a real-analytic function of time. The rate of convergence is derived explicitly, which depends on the function of surface heat flux. Numerical examples are provided to validate the convergence performance.