A collection of numerical simulations in my work.
A collection of numerical simulations in my work.
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Structure-preserving schemes and fast solvers for multiphysics systems
Structure-preserving schemes and fast solvers for multiphysics systems
Structure-preserving discretisations are numerical schemes that reflect important algebraic, geometric, or topological features of the underlying continuous problem. Examples of such structures include conservation laws, dissipation laws, the symplecticity of the flow map (a geometric property shared by all Hamiltonian systems), and physical constraints (such as the solenoidality of the magnetic field). Therefore, it is essential to preserve those structures for high-fidelity numerical simulations.
Designing an efficient solver for large-scale simulation and extremely physical testing problems (e.g. large Reynolds numbers) in a multiphysics problem is essential. The aim is to use efficient preconditioning techniques to design robust solvers which are independent of discretisation and physical parameters.
Magnetic relaxation
Magnetic relaxation
Magnetic relaxation refers to the process by which a magnetic field converges to an equilibrium. In our paper, we design a numerical scheme to preserve the helicity, an essential topological invariant ensuring the physical fidelity.
2D lid-driven cavity
2D lid-driven cavity
The MHD equations describe the dynamics of conducting fluid under the influence of a magnetic field. This system is extremely difficult to solve due to the strong coupling of electromagnetism and hydrodynamics. Therefore, it is important to design efficient solvers. The following results demonstrate the performance of a fast solver, which is based on augmented Lagrangian preconditioner and parameter-robust multigrid methods for an incompressible MHD system.
Re = Rem = 1
Re = Rem = 1000
(Nonlinear) Average KSP iteration number for 2D time-dependent lid-driven cavity problem.
Orszag-Tang vortex
The following simulation is the Orszag-Tang vortex problem, which can be used to test whether the numerical scheme satisfies the solenoidality condition (divB = 0).
Thermally coupled MHD system (Boussinesq approximation)
Thermally coupled MHD system (Boussinesq approximation)
A simulation of Rayleigh-Bénard convection. A Riesz-preconditioner is used to obtain a parameter-robust solver.
Island-coalescence
The island-coalescence problem is used to model magnetic reconnection processes in large aspect ratio tokamaks. For a strong magnetic field in the toroidal direction, the flow can be described in a two-dimensional setting by considering a cross section of the tokamak. The physical fidelity is measured by the reconnection rate. As the magnetic Reynolds number increases, this reconnection zone narros, resulting in a sharper, yet shorter, peak. In addition, a "sloshing" effect occurs, where the island bounces a little before fully merging into one. This yields a peak in the reconnection rate, whose height oscillates as the islands come together.
Double shear layer problem
Double shear layer problem
The following results are generated by a helicity- and energy-preserving finite element scheme. Only a high-order scheme can capture the correct behaviour. In this 2D example, the helicity is trivially 0, and our scheme preserves the correct energy evolution. We choose dt = 0.001, T = 2.0, nu = 0.0001, h = 1/128. The energy evolution figure is for nu = 0.1.
s = 1
s = 2
s = 3
Reduce the time step size to dt = 0.0001, nu = 0.0001, T = 1.2
s = 1
s = 2
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