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Abstract: We present a multirate method that is particularly suited for integrating thesystems of Ordinary Differential Equations ODEs that arise in step models ofsurface evolution. The surface of a crystal lattice, that is slightly miscutfrom a plane of symmetry, consists of a series of terraces separated by steps.Under the assumption of axisymmetry, the step radii satisfy a system of ODEsthat reflects the steps- response to step line tension and step-stepinteractions. Two main problems arise in the numerical solution of theseequations. First, the trajectory of the innermost step can become singular,resulting in a divergent step velocity. Second, when a step bunchinginstability arises, the motion of steps within a bunch becomes very stronglystable, resulting in -local stiffness-. The multirate method introduced in thispaper ensures that small time steps are taken for singular and locally stiffcomponents, while larger time steps are taken for the remaining ones. Specialconsideration is given to the construction of high order interpolants duringrun time which ensures fourth order accuracy of scheme for components of thesolution sufficiently far away from singular trajectories.

Autor: Pak-Wing Fok, Rodolfo R. Rosales


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