A collection of numerical simulations in my work.

Magnetohydrodynamics

Magnetic relaxation

2D lid-driven cavity

Orszag-Tang vortex

Thermally coupled MHD system (Boussinesq approximation)

Island-coalescence

High-order scheme

Double shear layer problem

Magnetohydrodynamics

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 for an open problem called the Parker problem to preserve the helicity, an essential topological invariant ensuring the physical fidelity.

For a long time, it has been verified that a structure-preserving scheme will provide a quantitative improvement of the numerical results. However, standard discretisation like Taylor-Hood for Stokes still provides reasonable results.

The Parker problem is a striking example of the significance of using structure-preserving schemes. It has a huge qualitative improvement: e.g. you can get the nontrivial steady state only if you preserve the helicity, otherwise, you will get nothing!

Hopf fibre: this magnetic configuration has non-zero helicity.

IsoHelix relaxation: this magnetic configuration has zero helicity.

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)

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 narrows, 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.

High-order scheme

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

Other areas

Obstacle problem

The obstacle problem is a type of variational inequality that describes situations where a solution must stay above a given obstacle while satisfying a differential equation. It is a key example of a free boundary problem, since the region where the solution touches the obstacle is not known beforehand.

Obstacle problems appear in many fields, such as elasticity, fluid flow, and optimal stopping. They are especially important in financial mathematics, where the pricing of American options can be modeled as an obstacle problem.

Phase-field

Phase-field models are usually constructed in order to reproduce a given interfacial dynamics. A number of formulations of the phase-field model are based on a free energy function. Thus, it is desirable to design numerical schemes that preserve the energy dissipation law. The picture shows the numerical solutions of the Cahn-Hilliard equations.

Solid mechanics

In materials science, geotechnical engineering, and biomechanics, we often encounter a special phenomenon: porous media that consist of a solid skeleton filled with fluid. The interaction between the solid and the fluid jointly determines the mechanical behavior of the material. The Biot model, proposed by Maurice A. Biot in 1941, is a mathematical framework that describes this solid–fluid coupling behavior.

At its core, the Biot model links the elastic deformation of the solid matrix with the flow of pore fluid, using partial differential equations to characterize the coupling among stress, strain, and pore pressure. It serves as a fundamental mathematical model in the field of poroelasticity.

The picture shows a 3D footing problem modeled by the 2-field Biot model. The Riesz preconditioner demonstrates robustness and efficiency, especially for nu=0.499, where the problem becomes superdifficult due to the material becoming incompressible.

linear iteration count of Riesz preconditioner

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