On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations

Meshfree solution schemes for the incompressible Navier--Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these algorithms that are specific to meshfree methods have often been overlooked. In this paper, we study the drawbacks of conventionally used meshfree Generalized Finite Difference Method~(GFDM) schemes for Lagrangian incompressible Navier-Stokes equations, both operator splitting schemes and monolithic schemes. The major drawback of most of these schemes is inaccurate local approximations to the mass conservation condition. Further, we propose a new modification of a commonly used monolithic scheme that overcomes these problems and shows a better approximation for the velocity divergence condition. We then perform a numerical comparison which shows the new monolithic scheme to be more accurate than existing schemes

Evaluation Of Glueball Masses From Supergravity

In the framework of the conjectured duality relation between large $N$ gauge theory and supergravity the spectra of masses in large $N$ gauge theory can be determined by solving certain eigenvalue problems in supergravity. In this paper we study the eigenmass problem given by Witten as a possible approximation for masses in QCD without supersymmetry. We place a particular emphasis on the treatment of the horizon and related boundary conditions. We construct exact expressions for the analytic expansions of the wave functions both at the horizon and at infinity and show that requiring smoothness at the horizon and normalizability gives a well defined eigenvalue problem. We show for example that there are no smooth solutions with vanishing derivative at the horizon. The mass eigenvalues up to $m^{2}=1000$ corresponding to smooth normalizable wave functions are presented. We comment on the relation of our work with the results found in a recent paper by Cs\'aki et al., hep-th/9806021, which addresses the same problem.Comment: 20 pages,Latex,3 figs,psfig.tex, added refs., minor change

Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions

On the convergence of local expansions of layer potentials

In a recently developed quadrature method (quadrature by expansion or QBX), it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on surface by making use of local expansions about carefully chosen off-surface points. In this paper, we derive estimates for the rate of convergence of these local expansions, providing the analytic foundation for the QBX method. The estimates may also be of mathematical interest, particularly for microlocal or asymptotic analysis in potential theory

Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs

We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of dominant principal components. The focus of the present article is the derivation of sharp error bounds for the constant coefficient case and a first and second order approximation. We give a precise characterisation when these bounds hold for (non-smooth) option pricing applications and provide numerical results demonstrating that the practically observed convergence speed is in agreement with the theoretical predictions