Utilizing wavelets to solve high-dimensional transport equations in nano-devices

In this work wavelets are considered for the deterministic solutions of 6-dimensional semiconductor transport problems for the first time. The advantages of wavelets over conventional approaches are often strongly amplified in higher dimensions by the construction of so-called sparse grids. The Multi-Wavelet Discontinuos Galerkin Method is derived which is not only flux conservative and stabilizing but, in contrast to other methods, also allows to use high-order piercewise polynomials and to construct unstructured meshes in all dimensions. Special adaptive criteria for transport problems are derived and demonstrated to work satisfactory for high polynomial order Multi-Wavelets in all phase space directions so that efficient adaptive sparse grids can automatically be found. Fully adaptive nano-device simulations demonstrate savings related to the sparse grids of more than 99%.