Narrow Results

A recent improvement of the lattice BGK model, based on a regularization of the precollision distribution function, is applied to three dimensional Womersley flow. The accuracy and the stability of the model are essentially improved by using this regularization. A good agreement with analytical Womersley solution is presented, as well as an improvement of the accuracy over standard L-BGK. Numerical stability of the scheme for a range of Reynolds and Womersley numbers is also presented, demonstrating an enhancement of the stability range of L-BGK for this type of flows.

We consider the correctness of 2-d Delaunay triangulation algorithms implemented using floating-point arithmetic. The α-pseudocircle through points a, b, c consists of three circular arcs connecting ab, bc, and ac, each arc inside the circumcircle of a, b, c and forming angle α with the circumcircle; a triangulation is α-empty if the α-pseudocircle through the vertices of each triangle is empty. We show that a simple Delaunay triangulation algorithm—the flipping algorithm—can be implemented to produce O(n∈)-empty triangulations, where n is the number of point sites and ∈ is the relative error of floating-point arithmetic; its worst-case running time is O(n²). We also discuss floating-point implementation of other 2-d Delaunay triangulation algorithms.

The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully nonlinearly coupled formulation for the study of fluid structure interactions.

Many numerical methods have been introduced to reduce the difficulties related to the nonlinear coupling between the structure and the fluid evolution. However numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates of the variational formulation of the immersed boundary method.

A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly.

We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.

The Immersed Boundary Method (IBM) has been designed by Peskin for the modeling and the numerical approximation of fluid-structure interaction problems, where flexible structures are immersed in a fluid. In this approach, the Navier–Stokes equations are considered everywhere and the presence of the structure is taken into account by means of a source term which depends on the unknown position of the structure. These equations are coupled with the condition that the structure moves at the same velocity of the underlying fluid.

Recently, a finite element version of the IBM has been developed, which offers interesting features for both the analysis of the problem under consideration and the robustness and flexibility of the numerical scheme. Initially, we considered structure and fluid with the same density, as it often happens when dealing with biological tissues. Here we study the case of a structure which can have a density higher than that of the fluid. The higher density of the structure is taken into account as an excess of Lagrangian mass located along the structure, and can be dealt with in a variational way in the finite element approach.

The numerical procedure to compute the solution is based on a semi-implicit scheme. In fluid-structure simulations, nonimplicit schemes often produce instabilities when the density of the structure is close to that of the fluid. This is not the case for the IBM approach. In fact, we show that the scheme enjoys the same stability properties as in the case of equal densities.

We study stability, dispersion and dissipation properties of four numerical schemes (Itera-tive Crank–Nicolson, 3rd and 4th order Runge–Kutta and Courant–Fredrichs–Levy Nonlinear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar nonlinear wave equation with a type of nonlinearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant–Fredrichs–Levy Nonlinear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4th order Runge–Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.

Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten.

We develop and study stability properties of a hybrid approximation of functionals of the Bates jump model with stochastic interest rate that uses a tree method in the direction of the volatility and the interest rate and a finite-difference approach in order to handle the underlying asset price process. We also propose hybrid simulations for the model, following a binomial tree in the direction of both the volatility and the interest rate, and a space-continuous approximation for the underlying asset price process coming from a Euler–Maruyama type scheme. We test our numerical schemes by computing European and American option prices.

A change in the representation of discrete motion equations for nonlinear structural dynamics of two-dimensional bodies is developed. The objective is to write the motion equation in a less nonlinear form. This leads to a significant increase in the range of stability of the time integration process and a reduction in the number of Newton iterations required in the time integration step.

A new family of explicit model-based integration algorithms for solving the equations of motion for linear and nonlinear systems is developed. These algorithms are also known as structure-dependent algorithms because the integration parameters are functions of the complete model of the structural system. A variety of numerical properties of the proposed algorithms, including consistency and local truncation error, stability, numerical dispersion and energy dissipation, overshooting, and frequency response under arbitrary excitation, are investigated using the discrete control theory and amplification matrix for linear elastic systems. In addition, the discrete control theory is applied for assessing the stability of the proposed algorithms for nonlinear structural systems. It is observed that the proposed algorithms exhibit the same numerical characteristics as the well-known Newmark family of integration algorithms. Compared with three existing model-based integration algorithms, i.e. the Chen–Ricles, modified Chen–Ricles, and Gui’s algorithms, the proposed algorithms possess more general and versatile numerical features. As a result, the new family of explicit model-based integration algorithms can be potentially used to solve complicated linear and nonlinear structural dynamics problems.

It is known that electrostatic potential (ESP) derived atomic charges for molecular systems suffer from rank deficiency in least-square fitting. In this paper, we studied numerical properties of atomic charges by fitting to ESP and electric field (EF) for a series of molecular systems, ranging from small molecules to peptides. Our study shows that although atomic charges of certain atoms in a molecule may differ a lot resulting from different fitting schemes, the effect on some observed physical properties of the molecules, such as solvation free energy and radial distributions of the solvent, are negligible. For all the molecules studied in this paper, it is shown that by incorporating EF terms in the least-square fitting, the rank of the least-squares matrix is increased, and the full rank is achieved when only EF terms are included. The current study demonstrates that by using EF, instead of the widely used ESP, in least-square fitting of atomic charges, one can obtain numerically more stable and better defined atomic charges in molecules. Such derived atomic charges may have more desirable properties and better numerical stabilities in studying detailed molecular processes such as in molecular dynamics simulation of macromolecules.

The generalized finite element method (GFEM) is applied to a nonconventional hybrid-mixed stress formulation (HMSF) for plane analysis. In the HMSF, three approximation fields are involved: stresses and displacements in the domain and displacement fields on the static boundary. The GFEM–HMSF shape functions are then generated by the product of a partition of unity associated to each field and the polynomials enrichment functions. In principle, the enrichment can be conducted independently over each of the HMSF approximation fields. However, stability and convergence features of the resulting numerical method can be affected mainly by spurious modes generated when enrichment is arbitrarily applied to the displacement fields. With the aim to efficiently explore the enrichment possibilities, an extension to GFEM–HMSF of the conventional Zienkiewicz-Patch-Test is proposed as a necessary condition to ensure numerical stability. Finally, once the extended Patch-Test is satisfied, some numerical analyses focusing on the selective enrichment over distorted meshes formed by bilinear quadrilateral finite elements are presented, thus showing the performance of the GFEM–HMSF combination.

Explicit algebraic expressions are derived for the roots of a cubic equation having one real and two complex roots. As opposed to traditional methods for direct (that is, non-iterative) solution, evaluation of the expressions is unconditionally numerically stable. A floating-point error analysis of the computations is given, which shows that the computed roots have extremely small relative errors. The computations are proved to be backward stable, that is, the computed roots are exact for a closely neighbouring cubic equation. Applications and uncertainty propagation are also considered.