Adaptive gradient-augmented level set method with multiresolution error estimation

A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Her-mite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quad-tree data structures. For adaptive time integration, an embedded Runge–Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported. 1. Introduction. In some advection dominated problems, the solution develops small-scale features but remains smooth during time evolution. These problems can be solved efficiently using numerical methods based on high-order interpolation on fixed Eulerian grids [21, 14]. If the small-scale features are localized in some part of the computational domain, its non-uniform partition with grid points clustered at the same part of the domain allows reducing the cost of computation without loosing accuracy. However, if the location of these features changes in time, the efficiency of the numerical method can be significantly improved by adapting the partition dynamically to the solution (see, e.g., [8] and references therein). When applied to pure advection problems, Eulerian schemes require some stabilization which introduces numerical diffusion and thus pollutes the solution. Another drawback are small time steps imposed by the stability limit of explicitly discretized Eulerian schemes. Semi-Lagrangian schemes combine advantages of Eulerian schemes, for example for solving Poisson equations on a grid, with those of Lagrangian schemes, especially that there is no time step restriction. A review on semi-Lagrangian schemes introduced in the context of numerical weather prediction can be found e.g. in [23]. These schemes have also been used in the context of plasma physics for solving the Vlasov equation, e.g. [22]. In this paper, we present an adaptive method for the two-dimensional advection equation based on a semi-Lagrangian approach. Advection problems are encountered for example in moving fronts for a given velocity field, or in transport of passive scalars modeling pollution or mixing in chemical engineering [18]. It can also be viewed as a simple model that partly describes other, more complex problems, such as advection-reaction-diffusion, fluid flow, elasticity, etc. Therefore the proposed numerical method may be relevant to those problems as well. We present a generalization of the gradient-augmented level set method [16] to adaptive discretization in …