T. Schiekofer and G. W. Zumbusch.
Software concepts of a sparse grid finite difference code.
In W. Hackbusch and G. Wittum, editors, Proceedings of the 14th
GAMM-Seminar Kiel on Concepts of Numerical Software, Notes on Numerical
Fluid Mechanics, page 11, Wiesbaden, Germany, 1998. Vieweg.
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Sparse grids provide an efficient representation of discrete solutions of PDEs and are mainly based on specific tensor products of one-dimensional hierarchical basis functions. They easily allow for adaptive refinement and compression. We present special finite difference operators on sparse grids that possess nearly the same properties as full grid operators. Using this approach, partial differential equations of second order can be discretized straightforwardly. We report on an adaptive finite difference research code implementing this on sparse grids. It is structured in an object oriented way. It is based on hash storage techniques as a new data structure for sparse grids. Due to the direct access of arbitrary data traditional tree like structures can be avoided. The above techniques are employed for the solution of parabolic problems. We present a simple space-time discretization. Furthermore a time-stepping procedure for the solution of the Navier Stokes equations in 3D is presented. Here we discretize by a projection method and obtain Poisson problems and convection-diffusion problems.