Sparse grid WENO schemes for high spatial dimension hyperbolic and convection-diffusionequations
Sparse grid WENO schemes for high spatial dimension hyperbolic and convection-diffusionequations
时间：6月3日 上午10:00――11:00
地点：工商楼200-9
报告人: Yongtao Zhang (University of Notre Dame)
摘要: In recent years, sparse grid techniques have been used broadly as an efficient approximation tool for high-dimensional problemsin many scientific and engineering applications. In this talk, I will present our recent results on designing sparse grid weighted essentially non-oscillatory (WENO) schemes for solving high spatial dimension convection-diffusion equations and hyperbolic PDEs. Our goal is to apply sparse grid techniques in high order schemes to achieve more efficient computations than that in their regular performance in solving multidimensional PDEs. A challenge is how to design the schemes on sparse grids such that comparable high order accuracy of the schemes in smooth regions of the solutions can still be achieved as that for computations on regular single grids. For problems with discontinuous solutions, additional challenge is that essentially non-oscillatory stability in non-smooth regions of the solutions needs to be preserved in the sparse grid schemes. We apply sparse-grid combination approach to overcome these difficulties. To deal with discontinuous solutions, we apply WENO interpolation for the prolongation part in sparse-grid combination techniques. Two dimensional (2D), three dimensional (3D) and four dimensional (4D) numerical examples with smooth or non-smooth solutions are presented. It is shown that significant computational times are saved especially for higher dimensional problems, while both accuracy and stability of the original schemes are maintained well for numerical simulations on refined sparse grids.