High order accuracy WENO methods for high dimensional problems on sparse grids

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

High order accuracy numerical methods are especially efficient for solving mathematical models in computational fluid dynamics and computational biology which contain complex solution structures. The computational cost increases significantly when the number of grid points is large or the spatial dimension of the problem is high, due to the "curse of dimensionality". How to achieve fast computations by high order accuracy methods is a very important question especially for long-time simulations. This research project aims to develop efficient high order accuracy numerical methods on sparse grids for high spatial dimensional problems. The new methods have the potential to be applied to a broader class of applications in quantum electronic systems, molecular motors, finance, collective cell motions in biology, gene regulatory network, etc. The PI will design, analyze and implement novel high order Krylov integration factor (IF) weighted essentially nonoscillatory (WENO) algorithms for solving hyperbolic or convection-diffusion partial differential equation (PDE) problems on sparse grids by using the sparse-grid combination technique to deal with the high dimensional challenge. The sparse-grid method is a powerful approximation tool for high dimensional problems. It has been successfully used in many scientific and engineering applications. Discretizations on sparse grids involve much fewer degrees of freedom than that on single grids. Efficient numerical simulations of these high dimensional systems will help in studying interesting biological questions in this area. The proposed research will contribute in the active area of dealing with the "curse of dimensionality". A suite of powerful computational tools for solving high dimensional nonlinear PDEs will be developed. These techniques are expected to make positive contributions to computer simulations of complicated phenomena in biological and physical systems. The proposed activity will also provide excellent training and education opportunities for both graduate and undergraduate students interested in research at the interface of mathematics, computation, and applications.