PDE Methods in Flow Simulation Post Processing Joachim Becker Tobias Preußer Martin Rumpf May 10, 2007 Abstract Vector field visualization is an important topic in scientific visualization. The aim is to graphically represent field data in an intuitively understandable and pre- cise way. Two novel methods are described which enable an easy perception of flow data. The texture transport method especially applies to timedependent veloc- ity fields. Lagrangian coordinates are computed solving the corresponding linear transport equations numerically. Choosing an appropriate texture on the reference frame the coordinate mapping can be applied as a suitable texture mapping. Alter- natively, the aligned diffusion methods serves as an appropriate scale space method for the visualization of complicated flow patterns. It is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retain- ing and enhancing edges. Here an initial noisy image is smoothed along stream- lines, whereas the image is sharpened in the orthogonal direction. The two meth- ods have in common that they are based on a continuous model and discretized only in the final implementational step. Therefore, many important properties are naturally established already in the continuous model. 1 Introduction The visualization of field data, especially of velocity fields from CFD computations is one of the fundamental tasks in scientific visualization. A variety of different ap- proaches has been presented. The simplest method to draw vector plots at nodes of some overlayed regular grid in general produces visual clutter, because of the typically different local scaling of the field in the spatial domain, which leads to disturbing mul- tiple overlaps in certain regions, whereas in other areas small structures such as eddies can not be resolved adequately. The central goal is to obtain a denser, intuitively bet- ter receptible method. Furthermore it should be closely related to the mathematical meaning of field data, which is mainly expressed in its one to one relation to the corre- sponding flow. If a vector field v :Ω × IR + 0 → IR n for some domain Ω ⊂ IR n is given, then the corresponding flow φ :Ω × IR + 0 → Ω is described by the system of ordinary differential equations ∂ t φ(x,t)= v(φ(x,t),t) 1