An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

Seminar: 
Applied Mathematics
Event time: 
Wednesday, January 25, 2023 - 1:00pm
Location: 
AKW 200
Speaker: 
Fredrik Fryklund
Speaker affiliation: 
NYU
Event description: 

Boundary integral methods are attractive for solving homogeneous elliptic
partial differential equations on complicated geometries, since they can
offer accurate solutions with a computational cost that is linear or close
to linear in the number of discretization points on the boundary of the
domain. However, these numerical methods are not straightforward to apply to
time-dependent equations, which often arise in science and engineering. We
address this problem with an integral equation-based solver for the
advection-diffusion equation on moving and deforming geometries in two space
dimensions. In this method, an adaptive high-order accurate time-stepping
scheme based on semi-implicit spectral deferred correction is applied. One
time-step then involves solving a sequence of non-homogeneous modified
Helmholtz equations, a method known as elliptic marching. Our solution
methodology utilizes several recently developed methods, including special
purpose quadrature, a function extension technique and a spectral Ewald
method for the modified Helmholtz kernel. Special care is also taken to
handle the time-dependent geometries. The numerical method is tested through
several numerical examples to demonstrate robustness, flexibility and
accuracy.