IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Sparse grid Krylov IIF and WENO schemes for high spatial dimension convection-diffusion and hyperbolic equations

  • Oct. 16, 2017
  • 3:15 p.m.
  • LeConte 312

Abstract

In recent years, sparse grid techniques have been used broadly as an efficient approximation tool for high-dimensional problems in many scientific and engineering applications. In this talk, I will present our recent results on designing sparse grid Krylov implicit integration factor (IIF) scheme for solving high spatial dimension convection-diffusion-reaction equations, and sparse grid weighted essentially non-oscillatory (WENO) scheme for solving 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 sparse grid WENO scheme, additional challenge is that essentially non-oscillatory stability in non-smooth regions of the solutions needs to be preserved. We apply sparse-grid combination approach to overcome these difficulties. To deal with discontinuous solutions of hyperbolic PDEs, we apply WENO interpolation for the prolongation part in sparse-grid combination techniques. Both two dimensional (2D) and three dimensional (3D) numerical examples with smooth or non-smooth solutions are presented to show that significant computational times are saved, while both accuracy and stability of the original schemes are maintained for numerical simulations on sparse grids.

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