Numerical Analysis and Scientific Computing Seminar | Adi Ditkowski, Block Finite Difference Schemes and Their Relations to Discontinuous Galerkin Methods

Zoom (Please contact ddelreyfernandez@uwaterloo.ca for meeting link) 

Speaker

Adi Ditkowski, Professor, Department of Applied Mathematics Tel Aviv University

Title

Block Finite Difference Schemes and Their Relations to Discontinuous Galerkin Methods

Abstract

Block Finite Difference methods (BFD) are Finite Difference (FD) methods in which the domain is divided into blocks, or cells, containing two or more grid points with a different stencil used for each grid point. Unlike the standard FD method, in which the same stencil is used for every grid point. Using this approach, we can design the scheme such that the leading term of the truncation error lies in a different subspace than the solution. Different dynamics can then be assigned to the error and the solution. Therefore, we can construct high-performance schemes.

In this talk, we present BFD for parabolic and hyperbolic equations in one and multiple dimensions. For the parabolic equations, we show that we can use the dissipation of the scheme to get a fifth-order convergence rate from a third-order truncation scheme. Furthermore, a six-order convergence rate can be obtained using a post-processing filter. We show that we can reduce the phase error for the hyperbolic equations using BFD schemes while maintaining the rest of the errors bounded in time. We also show these BFD schemes are highly accurate, nodal-based Discontinuous Galerkin Methods.

This work was done in collaboration with Anne Le Blanc and Chi-Wang Shu.