Adaptive mesh refinement for thin-film equations

An adaptive finite difference method is developed for a class of fully nonlinear time-dependent thin liquid film equations. Equations of the type h_t + f_y(h) = -ε³∇·(M(h) ∇Δh) arise in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the fluid film height. Enhanced accuracy for the method is attained by covering the front with a sequence of nested, progressively finer, rectangular grid patches that dynamically follow the front motion. Results of numerical experiments illustrate the accuracy, the efficiency, and the robustness of the method.