Numerical methods for large-scale, time-dependent partial differential equations

A survey of numerical methods for time dependent partial differential equations is presented. The emphasis is on practical applications to large scale problems. A discussion of new developments in high order methods and moving grids is given. The importance of boundary conditions is stressed for both internal and external flows. A description of implicit methods is presented including generalizations to multidimensions. Shocks, aerodynamics, meteorology, plasma physics and combustion applications are also briefly described

Simulating bubble nucleation in the electroweak phase transition

First order electroweak phase transitions (EWPTs) are an attractive area of research. This is mainly due to two reasons. First, they contain aspects that could help to explain the observed baryon asymmetry. Secondly, strong first order PTs could produce gravitational waves (GWs) that could be detectable by the Laser Interferometer Space Antenna (LISA), a future space-based GW detector. However, the electroweak PT in the Standard Model (SM) is not a first order transition but a crossover. In so-called beyond the SM theories the first order transitions are possible. To investigate the possibility of an EWPT and the detection by LISA, we must be able to parametrise the nature of the PT accurately. We are interested in the calculation of the bubble nucleation rate because it can be used to estimate the properties of the possible GW signal, such as the duration of the PT. The nucleation rate essentially quantifies how likely it is for a point in space to tunnel from one phase to the other. The calculation can be done either using perturbation theory or simulations. Perturbative approaches however suffer from the so-called infrared problem and are not free of theoretical uncertainty. We need to perform a nonperturbative calculation so that we can determine the nucleation rate accurately and test the results of perturbation theory. In this thesis, we will explain the steps that go into a nonperturbative calculation of the bubble nucleation rate. We perform the calculation on the cubic anisotropy model, a theory with two scalar fields. This toy model is one of the simplest in which a radiatively induced transition occurs. We present preliminary results on the nucleation rate and compare it with the thin-wall approximation

Astro-GRIPS, the General Relativistic Implicit Parallel Solver for Astrophysical Fluid Flows

In this work the development of the simulation code Astro-GRIPS, the General Relativistic Implicit Parallel Solver, is performed, which solves the three-dimensional axi-symmetric general relativistic hydrodynamic Euler or Navier-Stokes equations under the assumption of a fixed background metric of a Schwarzschild or Kerr black hole using time-implicit methods. It is an almost total re-write of an old spaghetti-code like serial Fortran 77 simulation program. By modernization and optimization it is now a modern, well structured, user-friendly, flexible and extensible simulation program written in Fortran 90/95. The finite volume discretization ensures conservation and the defect-correction iteration strategy is used to resolve the non-linearities of the equations. One can use a variety of solution procedures that range from purely explicit up to fully implicit schemes with up to third order spatial and second order temporal accuracy. The large sparse linear equation systems used for the implicit methods can be solved by the Black-White Line-Gauß-Seidel relaxation method (BW-LGS), the Approximate Factorization Method (AFM) or by Krylov Subspace Iterative methods like GMRES. The optimal solution method and the coupling of equations is problem-dependent. Optimizations in the matrix construction, the MPI-Parallelization for distributed memory machines and several Newtonian and relativistic tests were conducted successfully

High Speed Switching in Magnetic Recording Thin-Film Heads

There has always been an increasing demand for high density data storage. However, the increased areal storage densities of hard disk drives require a level of miniturisation of the recording heads where the micromagnetic details and switching mechanisms can no longer be ignored. Furthermore, theoretical and numerical studies on thin-film recording heads tend to separate the micromagnetics from the electromagnetic aspects of the head during switching and hence ignore the lossy nature of head materials. This project was aimed to develop a numerical simulation approach that simultaneously incorporates the fundamental micromagnetic and electromagnetic details of magnetic materials to study the fast switching process in soft magnetic materials in general, and in thin-film inductive writers in particular. The project also was aimed at establishing an impedance measurement system to characterise losses in magnetic recording heads, and to allow comparison with the simulations. This project successfully met all its original objectives. A numerical technique to simulate the dynamic behaviour of magnetic materials and devices has been developed, and applied to study the switching process in thin-film recording heads. Two-dimensional simulations of complete commercial head structures including the coils and pole regions were carried out and parameters such as gap field rise times, gap field distributions, and core inductances, which are important for head designers, were predicted. Moreover, the role of eddy currents delaying the magnetisation switching was elucidated. Furthermore, it was found that the gradient of the recording fields were sharper near the conrner regions of the poles when considering magnetic details. A high precision, high bandwidth impedance measurement system was established to characterise losses in magnetic heads. Fittings of measured core inductances to a proposed equivalent circuit model of the core’s relaxation processes revealed the switching times of heads (of the order 0.1 to 1.0ns)

Subgrid scale modelling of transport processes.

Consideration of stabilisation techniques is essential in the development of physical models if they are to faithfully represent processes over a wide range of scales. Careful application of these techniques can significantly increase flexibility of models, allowing the computational meshes used to discretise the underlying partial differential equations to become highly nonuniform and anisotropic, for example. This exibility enables a model to capture a wider range of phenomena and thus reduce the number of parameterisations required, bringing a physically more realistic solution. The next generation of fluid flow and radiation transport models employ unstructured meshes and anisotropic adaptive methods to gain a greater degree of flexibility. However these can introduce erroneous artefacts into the solution when, for example, a process becomes unresolvable due to an adaptive mesh change or advection into a coarser region of mesh in the domain. The suppression of these effects, caused by spatial and temporal variations in mesh size, is one of the key roles stabilisation can play. This thesis introduces new explicit and implicit stabilisation methods that have been developed for application in fluid and radiation transport modelling. With a focus on a consistent residual-free approach, two new frameworks for the development of implicit methods are presented. The first generates a family of higher-order Petrov-Galerkin methods, and the example developed is compared to standard schemes such as streamline upwind Petrov-Galerkin and Galerkin least squares in accurate modelling of tracer transport. The dissipation generated by this method forms the basis for a new explicit fourth-order subfilter scale eddy viscosity model for large eddy simulation. Dissipation focused more sharply on unresolved scales is shown to give improved results over standard turbulence models. The second, the inner element method, is derived from subgrid scale modelling concepts and, like the variational multiscale method and bubble enrichment techniques, explicitly aims to capture the important under-resolved fine scale information. It brings key advantages to the solution of the Navier-Stokes equations including the use of usually unstable velocity-pressure element pairs, a fully consistent mass matrix without the increase in degrees of freedom associated with discontinuous Galerkin methods and also avoids pressure filtering. All of which act to increase the flexibility and accuracy of a model. Supporting results are presented from an application of the methods to a wide range of problems, from simple one-dimensional examples to tracer and momentum transport in simulations such as the idealised Stommel gyre, the lid-driven cavity, lock-exchange, gravity current and backward-facing step. Significant accuracy improvements are demonstrated in challenging radiation transport benchmarks, such as advection across void regions, the scattering Maynard problem and demanding source-absorption cases. Evolution of a free surface is also investigated in the sloshing tank, transport of an equatorial Rossby soliton, wave propagation on an aquaplanet and tidal simulation of the Mediterranean Sea and global ocean. In combination with adaptive methods, stabilising techniques are key to the development of next generation models. In particular these ideas are critical in achieving the aim of extending models, such as the Imperial College Ocean Model, to the global scale

Finite-Volume Filtering in Large-Eddy Simulations Using a Minimum-Dissipation Model

Large-eddy simulation (LES) seeks to predict the dynamics of the larger eddies in turbulent flow by applying a spatial filter to the Navier-Stokes equations and by modeling the unclosed terms resulting from the convective non-linearity. Thus the (explicit) calculation of all small-scale turbulence can be avoided. This paper is about LES-models that truncate the small scales of motion for which numerical resolution is not available by making sure that they do not get energy from the larger, resolved, eddies. To identify the resolved eddies, we apply Schumann’s filter to the (incompressible) Navier-Stokes equations, that is the turbulent velocity field is filtered as in a finite-volume method. The spatial discretization effectively act as a filter; hence we define the resolved eddies for a finite-volume discretization. The interpolation rule for approximating the convective flux through the faces of the finite volumes determines the smallest resolved length scale δ. The resolved length δ is twice as large as the grid spacing h for an usual interpolation rule. Thus, the resolved scales are defined with the help of box filter having diameter δ= 2 h. The closure model is to be chosen such that the solution of the resulting LES-equations is confined to length scales that have at least the size δ. This condition is worked out with the help of Poincarés inequality to determine the amount of dissipation that is to be generated by the closure model in order to counterbalance the nonlinear production of too small, unresolved scales. The procedure is applied to an eddy-viscosity model using a uniform mesh