Abstract

The simulation of time-dependent physical problems, such as flows of some kind, places high demands on the domain discretization in order to obtain high accuracy of the numerical solution. We present a moving mesh method in which the mesh points automatically move towards regions where high spatial resolution is required.
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The number of mesh points remains constant, but the local resolution increases by several orders of magnitude. The main contribution is the automatic balancing of adaptation criteria, such that the developed solver is robust without the need for manual fine-tuning of parameters. Moving mesh methods are adaptive methods also known as r-refinement. They use a monitor function that expresses the relative `importance' of adaptivity for each point in the domain. The equidistribution of these monitor values refines the mesh in important areas and slightly coarsens the mesh in areas that need less spatial resolution. This is opposed to h-refinement, where mesh cells can be recursively split into smaller cells (so there, the number of mesh cells is not constant). We include performance comparisons between our r-refinement method and h-refinement results. It is easy to obtain basic adaptivity, but in order to obtain truly powerful adaptivity there are two important aspects to monitor functions. Firstly, they should be able to detect all relevant features in the flow solution. In steady, simple solutions, local error measures are often used as monitor components. In our time-dependent nonlinear systems of partial differential equations, though, solution gradients (e.g., of the density) and derived physical quantities (e.g. entropy, vorticity, magnetic current) are better at decreasing the global error by mesh adaptation. Secondly, whatever monitor components are used, it is important to balance them automatically, such that the mesh movement is smooth in space and time. Our automatic balancing avoids the need for manual parameter fine-tuning by the user and makes the overall solver very robust for a wide range of problems. The algorithms and actual solvers in our work are designed for the class of nonlinear systems of hyperbolic partial differential equations in one and two dimensions. This includes linear advection problems, Burgers' equation, compressible gas dynamics and ideal magnetohydrodynamics (MHD). We combine the mesh movement method with a finite volume solver for the physical model PDEs. The two parts are decoupled, which is elegant and keeps the overall solver flexible. The finite volume method uses MUSCL-type upwinding and slope limiting and is equipped with the HLLC approximate Riemann solver for gas dynamics. The time-integration uses a second-order explicit predictor--corrector scheme. The automatically balanced mesh adaptation captures both strong and subtle flow features, including unforeseen physical phenomena, such as physical staircasing in 1D MHD. In 2D HD physical instabilities, such as Kelvin--Helmholtz are captured in great detail. We also apply the moving mesh method to 2D MHD problems, where the magnetic field remains divergence-free. This work also includes an extensive overview of practically all possible approaches for moving mesh adaptivity. We list numerous methods with their developments over the years and discuss their differences and similarities.
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