Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation
Abstract
The main objectives of this project is to develop efficient and accurate spectral methods for fractional PDEs and high-dimensional problems. Fractional partial differential equations (FPDEs) appear in the investigation of transport dynamics in complex systems which are governed by the anomalous diffusion and non-exponential relaxation patterns. The main difficulties for dealing with fractional differential equations are: (i) fractional derivatives are non-local operators; (ii) fractional PDEs in space are usually derived in unbounded domains and their solutions exhibit slow algebraic decay at infinity; (iii) when truncated to finite domains, fraction derivatives involve singular weight functions and the solutions of FPDEs are usually singular near the boundary. We developed several efficient approaches to deal with these problems. The main difficulty for solving high-dimensional problems is how to break the curse of dimensionality. We continued our work in developing fast sparse spectral methods for solving a class of moderately high dimensional elliptic equations in bounded and unbounded domains. Many complex nonlinear systems can be described as gradient flows. We developed a class of extremely efficient numerical methods for solving a large class of gradient flows.
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 09, 2019
- Accession Number
- AD1085893
Entities
People
- Jie Shen
Organizations
- Purdue University