Anna-Karin Tornberg

Regularization Techniques for Numerical Approximation of PDEs with Singularities

The rate of convergence for numerical methods approximating differential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coefficients. We shall discuss the technique of local regularization for handling these problems. New numerical methods are presented and analyzed and numerical examples are given. Some serious deficiencies in existing regularization methods are also pointed out.

A fast multipole method for the three-dimensional Stokes equations

Many problems in Stokes flow (and linear elasticity) require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions. In the fluid mechanics setting, these are typically the Stokeslet (the kernel of the single layer potential) or the Stresslet (the kernel of the double layer potential). In this paper, we present a simple and efficient method for the rapid evaluation of such fields, using a decomposition into a small number of Coulombic N-body problems, following an approach similar to that of Fu and Rodin Y. Fu, G.J. Rodin, Fast solution methods for three-dimensional Stokesian many-particle problems, Commun. Numer. Meth. En. 16 (2000) 145-149]. While any fast summation algorithm for Coulombic interactions can be employed, we present numerical results from a scheme based on the most modern version of the fast multipole method H. Cheng, L. Greengard, V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys. 155 (1999) 468-498]. This approach should be of value in both the solution of boundary integral equations and multiparticle dynamics.

Multi-dimensional quadrature of singular and discontinuous functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.

Spectrally accurate fast summation for periodic Stokes potentials

A spectrally accurate method for the fast evaluation of N-particle sums of the periodic Stokeslet is presented. Two different decomposition methods, leading to one sum in real space and one in reciprocal space, are considered. An FFT based method is applied to the reciprocal part of the sum, invoking the equivalence of multiplications in reciprocal space to convolutions in real space, thus using convolutions with a Gaussian function to place the point sources on a grid. Due to the spectral accuracy of the method, the grid size needed is low and also in practice, for a fixed domain size, independent of N. The leading cost, which is linear in N, arises from the to-grid and from-grid operations. Combining this FFT based method for the reciprocal sum with the direct evaluation of the real space sum, a spectrally accurate algorithm with a total complexity of O(NlogN) is obtained. This has been shown numerically as the system is scaled up at constant density.

A numerical method for simulations of rigid fiber suspensions

In this paper, we present a numerical method designed to simulate the challenging problem of the dynamics of slender fibers immersed in an incompressible fluid. Specifically, we consider microscopic, rigid fibers, that sediment due to gravity. Such fibers make up the micro-structure of many suspensions for which the macroscopic dynamics are not well understood. Our numerical algorithm is based on a non-local slender body approximation that yields a system of coupled integral equations, relating the forces exerted on the fibers to their velocities, which takes into account the hydrodynamic interactions of the fluid and the fibers. The system is closed by imposing the constraints of rigid body motions. The fact that the fibers are straight have been further exploited in the design of the numerical method, expanding the force on Legendre polynomials to take advantage of the specific mathematical structure of a finite-part integral operator, as well as introducing analytical quadrature in a manner possible only for straight fibers. We have carefully treated issues of accuracy, and present convergence results for all numerical parameters before we finally discuss the results from simulations including a larger number of fibers.

Delta function approximations in level set methods by distance function extension

In [A.-K. Tornberg, B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys. 200 (2004) 462-488], it was shown for simple examples that the then m ...

Spectral accuracy in fast Ewald-based methods for particle simulations

A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.

An embedded boundary method for soluble surfactants with interface tracking for two-phase flows

Surfactants, surface reacting agents, lower the surface tension of the interface between fluids in multiphase flow. This capability of surfactants makes them ideal for many applications, including wetting, foaming, and dispersing. Due to their molecular composition, surfactants are adsorbed from the bulk fluid to the interface between the fluids, leading to different concentrations on the interface and in the fluid. In a previous paper [21], we introduced a new second order method using uniform grids to simulate insoluble surfactants in multiphase flow. This method used Strang splitting allowing for a fully second order treatment in time. Here, we use the same numerical methods to explicitly represent the singular interface, treat the interfacial surfactant concentration, and couple with the Navier-Stokes equations. Now, we introduce a second order method for the surfactants in the bulk that continues to allow the use of regular grids for the full problem. Difficulties arise since the boundary condition for the bulk concentration, which handles the flux of surfactant between the interface and bulk fluid, is applied at the interface which cuts arbitrarily through the regular grid. We extend the embedded boundary method, introduced in [22], to handle this challenge. Through our results, we present the effect of the solubility of the surfactants. We show results of drop dynamics due to resulting Marangoni stresses and of drop deformations in shear flow in the presence of soluble surfactants. There is a large nondimensional parameter space over which we try to understand the drop dynamics.

Consistent boundary conditions for the Yee scheme

A new set of consistent boundary conditions for Yee scheme approximations of wave equations in two space dimensions are developed and analyzed. We show how the classical staircase boundary conditions for hard reflections or, in the electromagnetic case, conducting surfaces in certain cases give O(1) errors. The proposed conditions keep the structure of the Yee scheme and are thus well suited for high performance computing. The higher accuracy is achieved by modifying the coefficients in the difference stencils near the boundary. This generalizes our earlier results with Gustafsson and Wahlund in one space dimension. We study stability and convergence and we present numerical examples.

Gravity induced sedimentation of slender fibers

Gravity induced sedimentation of slender, rigid fibers in a highly viscous fluid is investigated by large scale numerical simulations. The fiber suspension is considered on a microscopic level and the flow is described by the Stokes equations in a three dimensional periodic domain. Numerical simulations are performed to study in great detail the complex dynamics of a cluster of fibers. A repeating cycle is identified. It consists of two main phases: a densification phase, where the cluster densifies and grows, and a coarsening phase, during which the cluster becomes smaller and less dense. The dynamics of these phases and their relation to fluctuations in the sedimentation velocity are analyzed. Data from the simulations are also used to investigate how average fiber orientations and sedimentation velocities are influenced by the microstructure in the suspension. The dynamical behavior of the fiber suspension is very sensitive to small random differences in the initial configuration and a number of realizations of each numerical experiment are performed. Ensemble averages of the sedimentation velocity and fiber orientation are presented for different values of the effective concentration of fibers and the results are compared to experimental data. The numerical code is parallelized using the Message Passing Instructions (MPI) library and numerical simulations with up 800 fibers can be run for very long times which is crucial to reach steady levels of the averaged quantities. The influence of the periodic boundary conditions on the process is also carefully investigated.