An integral equation method for the advection-diffusion equation on time-dependent domains in the plane
arXiv:2206.08825 · doi:10.1016/j.jcp.2022.111856
Abstract
Boundary integral methods are attractive for solving homogeneous linear constant coefficient elliptic partial differential equations on complex 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
References in corpus (14)
- Fast convolution with free-space Green's functions
- A fast integral equation method for solid particles in viscous flow using quadrature by expansion
- Fast integral equation methods for the modified Helmholtz equation
- Efficient and accurate solver of the three-dimensional screened and unscreened Poisson's equation with generic boundary conditions
- A cut finite element method for incompressible two-phase Navier-Stokes flows
- A highly accurate boundary integral equation method for surfactant-laden drops in 3D
- Fast Ewald summation for free-space Stokes potentials
- An accurate integral equation method for simulating multi-phase Stokes flow
- A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the Immersed Boundary Smooth Extension method
- Partition of Unity Extension of Functions on Complex Domains
- A Fast Integral Equation Method for the Two-Dimensional Navier-Stokes Equations
- Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow
- An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains
- An integral equation method for closely interacting surfactant-covered droplets in wall-confined Stokes flow