Abstract
This paper presents
a high order method for solving the unbounded Poisson equation on a regular mesh using a Green’s function
solution. The high order convergence was achieved by formulating mollified integration kernels, that were derived from a filter
regularisation of the solution field. The method was implemented on a rectangular domain using fast Fourier transforms (FFT) to
increase computational
efficiency. The Poisson solver was extended to directly solve the derivatives of the solution. This is achieved
either
by including the differential operator in the integration kernel or by performing the differentiation as a multiplication of the
Fourier coefficients. In this way, differential operators such as the divergence or curl of the solution field could be solved to the
same
high order convergence without additional computational effort. The method was applied and validated using the equations
of
fluid mechanics as an example, but can be used in many physical problems to solve the Poisson equation on a rectangular
unbounded domain.
For the two-dimensional case we propose an infinitely smooth test function which allows for arbitrary high
order convergence. Using Gaussian smoothing as regularisation we document an increased convergence rate up to tenth order.
The method
however, can easily be extended well beyond the tenth order. To show the full extend of the method we present the
special case
of a spectrally ideal regularisation of the velocity formulated integration kernel, which achieves an optimal rate of
convergence.
Original language | English |
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Journal | I U T A M. Procedia |
Volume | 18 |
Pages (from-to) | 56-65 |
ISSN | 2210-9838 |
DOIs | |
Publication status | Published - 2015 |
Bibliographical note
This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords
- Poisson solver
- Elliptic solver
- Unbounded domain
- Infinite domain
- Isolated system
- Green’s function solution
- Numerical integration;
- Vortex methods
- Particle-mesh methods