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We study the performace of adaptive spline interpolation in semi--Lagrangian discretization schemes for Hamilton--Jacobi--Bellman equations. We investigate the local approximation properties of cubic splines on locally refined grids by a theoretical analysis. Numerical examples show how this method performs in practice. Using those examples we also illustrate numerical stability issues.doi:10.15495/epub_ubt_00005527 fatcat:tcgsxahiavfixef6jgdqmrls4q