FFT for Magnetrohydrodynamic Simulations
Benson Muite
Kichakato Kizito
Kenya
Outline
- Introduction
- Areas of investigation
- Equations
- Computational requirements
- References
Introduction
- FFT enables accurate simulations
- FFT is communication intensive, not great for modern parallel architectures
- Spectra useful in understanding behavior of the differential equation
- FFT can give good preservation of geometric properties of discretized equations
Introduction
- Examine incompressible case
- Model equations to understand reduced solar physics
- Also of mathematical interest
Areas of Interest
- Magnetohydrodynamic turbulence with and without viscosity
- Extend computational Euler investigations to a new setting - magnetohydrodynamics without viscosity
Equations
\[\begin{aligned}
& \frac{\partial u}{\partial x} \rightarrow ik_x\hat{u}
\end{aligned}\]
Equations
\[\begin{aligned}
& \frac{\partial\mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} = -\nabla p + \mathbf{b}\cdot\nabla\mathbf{b}-\nabla(\mathbf{b}\cdot\mathbf{b})+\Delta\mathbf{u} \\
& \frac{\partial\mathbf{b}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{b} = \mathbf{b}\cdot\nabla\mathbf{u} + \Delta \mathbf{b} \\
& \nabla\cdot\mathbf{u} = 0 \\
& \nabla\cdot\mathbf{b} = 0
\end{aligned}\]
Equations - no viscosity
\[\begin{aligned}\require{cancel}
& \frac{\partial\mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} = -\nabla p + \mathbf{b}\cdot\nabla\mathbf{b}-\nabla(\mathbf{b}\cdot\mathbf{b})\xcancel{+\Delta\mathbf{u}} \\
& \frac{\partial\mathbf{b}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{b} = \mathbf{b}\cdot\nabla\mathbf{u} \xcancel{+ \Delta \mathbf{b}} \\
& \nabla\cdot\mathbf{u} = 0 \\
& \nabla\cdot\mathbf{b} = 0
\end{aligned}\]
Discretization - 2D
\[\begin{aligned}
& u = \frac{\partial\psi}{\partial y} \quad v = - \frac{\partial\psi}{\partial x} \\
& b_x = \frac{\partial\phi}{\partial y} \quad b_y = -\frac{\partial\phi}{\partial x} \\
& \omega = \nabla \times \mathbf{u} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = - \Delta \psi \\
& \alpha = \nabla \times \mathbf{b} = \frac{\partial b_y}{\partial x} - \frac{\partial b_x}{\partial y} = -\Delta \phi
\end{aligned}\]
Discretization - 2D
\[\begin{aligned}
& \frac{\partial\omega}{\partial t} + u\frac{\partial \omega}{\partial x} + v\frac{\partial\omega}{\partial y} = b_x\frac{\partial \alpha}{\partial x} + b_y\frac{\partial \alpha}{\partial y} + \Delta \omega \\
& \frac{\partial\alpha}{\partial t} + u\frac{\partial\alpha}{\partial x} + v\frac{\partial\alpha}{\partial y} = b_x\frac{\partial \omega}{\partial x} + b_y\frac{\partial \omega}{\partial y} + \Delta \alpha \\
& \Delta \psi = -\omega \\
& \Delta \phi = -\alpha
\end{aligned} \]
Discretization - 3D
\[\begin{aligned}
& \frac{\partial \mathbf{u}}{\partial t} = \Delta \mathbf{u} + \mathbf{b}\cdot\nabla\mathbf{b} - \mathbf{u}\cdot\nabla\mathbf{u} - \nabla(\mathbf{b}\cdot\mathbf{b}) \\
& \quad + \nabla\Delta^{-1}[\nabla\cdot(\mathbf{u}\cdot\nabla\mathbf{u})-(\mathbf{b}\cdot\nabla\mathbf{b})] \\
& \frac{\partial \mathbf{b}}{\partial t} = \Delta \mathbf{b} + \mathbf{b}\cdot\nabla\mathbf{u} - \mathbf{u}\cdot\nabla{b} \\
& \quad + \nabla\Delta^{-1}[\nabla\cdot(\mathbf{u}\cdot\nabla\mathbf{b})-(\mathbf{b}\cdot\nabla\mathbf{u})]
\end{aligned}\]
Discretization - 3D, no viscosity
\[\begin{aligned}\require{cancel}
& \frac{\partial \mathbf{u}}{\partial t} =\xcancel{ \Delta \mathbf{u} +} \mathbf{b}\cdot\nabla\mathbf{b} - \mathbf{u}\cdot\nabla\mathbf{u} - \nabla(\mathbf{b}\cdot\mathbf{b}) \\
&\quad + \nabla\Delta^{-1}[\nabla\cdot(\mathbf{u}\cdot\nabla\mathbf{u})-(\mathbf{b}\cdot\nabla\mathbf{b})] \\
& \frac{\partial \mathbf{b}}{\partial t} = \xcancel{ \Delta \mathbf{b} +} \mathbf{b}\cdot\nabla\mathbf{u} - \mathbf{u}\cdot\nabla{b} \\
&\quad + \nabla\Delta^{-1}[\nabla\cdot(\mathbf{u}\cdot\nabla\mathbf{b})-(\mathbf{b}\cdot\nabla\mathbf{u})]
\end{aligned}\]
Computational requirements
- 2D - 9 scalar fields x 3/4/5 for time discretization + 2 fields for nonlinear terms, 655362 grid points
- 3D - 30 scalar fields x 3/4/5 for time discretization + 6 fields for nonlinear terms, 163843 grid points
- At high resolution quadruple precision can be helpful
- Calculate \[\lVert (i\mathbf{k})^p\hat{\mathbf{u}} \rVert \]
References
- https://en.wikibooks.org/wiki/Parallel_Spectral_Numerical_Methods/Incompressible_Magnetohydrodynamics
- Brachet, M.E.; Bustamante, M.D., Krstulovic, G.; Mininni, P.D.; Pouquet, A.; Rosenberg. (2013). "Ideal evolution of magnetohydrodynamics turbulence when imposing Taylor-Green symmetries. Physical Review E 87, 013110.
- Germain, P.; Ibrahim, S.; Masmoudi, N. (2012). "Well Posedness of the Navier-Stokes-Maxwell Equations". arXiv pre-print.
References
- Orszag, S.; Tang, C.M. (1979). "Small-scale structure of two-dimensional magnetohydrodynamic turbulence". Journal of Fluid Mechanics 90 (1): 129-143.
- Verma, Mahendra (2004). "Statistical theory of magnetohydrodynamic turbulence: recent results". Physics Reports 401: 220-380.
Acknowledgements
- University of Michigan
- King Abdullah University of Science and Technology
- University of Tartu
- Samar Aseeri