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To obtain a linear scaling law for the computation time the narrow band method was introduced [3]. This approach makes use of the fact that only the LS values of grid points close to the surface have an influence on the zero level set and thus on the actual position of the surface. By updating only a subset of grid points in time, namely only those within a band around the surface with a typical width of a couple of grid spacings, the computation time can be drastically reduced. Since the number of these so-called active grid points within the band is approximately the surface area times the narrow band width, an optimal linear scaling for the time integration is obtained. From time to time, the narrow band needs to be re-initialized, if the surface approaches its boundary.

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Otmar Ertl: Numerical Methods for Topography Simulation