Yiorgos-Sokratis Smyrlis

Some Aspects of the Method of Fundamental Solutions for Certain Harmonic Problems

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. The basic ideas of the MFS were introduced by Kupradze and Alexidze and its modern form was proposed by Mathon and Johnston. In this work, we investigate certain aspects of a particular version of the MFS, also known as the Charge Simulation Method, when it is applied to the Dirichlet problem for Laplaces equation in a disk.

A SINGULAR FUNCTION BOUNDARY INTEGRAL METHOD FOR THE LAPLACE EQUATION

The authors present a new singular function boundary integral method for the numerical solution of problems with singularities which is based on approximation of the solution by the leading terms of the local asymptotic expansion. The essential boundary conditions are weakly enforced by means of appropriate Lagrange multipliers. The method is applied to a benchmark Laplace-equation problem, the Motz problem, giving extremely accurate estimates for the leading singular coefficients. The method converges exponentially with the number of singular functions and requires a low computational cost. Comparisons are made to the analytical solution and other numerical methods.

The route to chaos for the Kuramoto-Sivashinsky equation

We present the results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. Our concern is with the asymptotic nonlinear dynamics as the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic (in time) attractor, to another steady fully modal attractor, to another time-periodic attractor, to a steady tetramodal attractor, to another time-periodic attractor having a full sequence of period-doublings (in the parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the lengths of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period-doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaums universal constant of 4.669201609.... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Applicability and applications of the method of fundamental solutions

In the present work, we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems. More specifically, we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration. In our study, the singularities of the fundamental solutions lie on a prescribed pseudo―boundary ― the boundary of a domain which embraces the domain of the problem under consideration. We extend previous density results of Kupradze and Aleksidze, and of Bogomolny, to more general domains and partial differential operators, and with respect to more appropriate norms. Our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement, namely the segment condition. Our density results are with respect to the norms of the spaces C l (Ω). Analogous density results are obtainable with respect to Holder norms. We have studied approximation by fundamental solutions of the Laplacian, biharmonic and m―harmonic and modified Helmholtz and poly-Helmholtz operators. In the case of elliptic systems, we obtain analogous density results for the Cauchy―Navier operator as well as for an operator which arises in the linear theory of thermo―elasticity. We also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space. Finally, we show that linear combinations of fundamental solutions of operators of order m > 4, with singularities lying on a prescribed pseudo-boundary, are not in general dense in the corresponding solution space.

Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients

In this paper the nonlinear stability of two-phase core-annular flow in a pipe is examined when the acting pressure gradient is modulated by time harmonic oscillations and viscosity stratification and interfacial tension is present. An exact solution of the Navier-Stokes equations is used as the background state to develop an asymptotic theory valid for thin annular layers, which leads to a novel nonlinear equation describing the spatio-temporal evolution of the interface. The evolution equation is an extension of the equation found for constant pressure gradients and generalizes the Kuramoto-Sivashinsky equation with dispersive effects found by Papageorgiou, Maldarelli and Rumschitzki, Phys. Fluids A2(3), 340–352 (1990), to a similar system with time periodic coefficients. The distinct regimes of slow and moderate flow are considered and the corresponding evolution is derived. Certain solutions are described analytically in the neighborhood of the first bifurcation point by use of multiple scales asymptotics. Extensive numerical experiments, using dynamical systems ideas, are carried out in order to evaluate the effect of the oscillatory pressure gradient on the solutions in the presence of a constant pressure gradient.

A matrix decomposition MFS algorithm for axisymmetric potential problems

The method of fundamental solutions is a boundary-type meshless method for the solution of certain elliptic boundary value problems. By exploiting the structure of the matrices appearing when this method is applied to certain three-dimensional potential problems, we develop an efficient matrix decomposition algorithm for their solution.

A Linear Least-Squares MFS for Certain Elliptic Problems

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we propose an efficient algorithm for the linear least-squares version of the MFS, when applied to the Dirichlet problem for certain second order elliptic equations in a disk. Various aspects of the method are discussed and a comparison with the standard MFS is carried out. Numerical results are presented.

Computational Study of the Dispersively Modified Kuramoto-Sivashinsky Equation

We analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified Kuramoto-Sivashinsky equations. Time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. The general case analyzed covers several physical applications arising in multiphase hydrodynamics and the emerging dynamics arise from a competition of long-wave instability (negative diffusion), short-wave damping (fourth order stabilization), nonlinear saturation (Burgers nonlinearity), and dispersive effects. The solutions of such systems typically converge to compact absorbing sets of finite dimension (i.e., global attractors) and are characterized by chaotic behavior. Our objective is to employ schemes which capture faithfully these chaotic dynamics. In the general case the dispersive term is taken to be a pseudodifferential operator which is allowed to have higher order than the familiar fourth order stabilizing term in the Kuramoto-Sivashinsky equation. In such instances we show that first and second order time-stepping schemes are appropriate and provide convergence proofs for the schemes. In physical situations when the dispersion is of lower order than the fourth order stabilization term (for example, a hybrid Kuramoto-Sivashinsky-Korteweg-deVries equation also known as the Kawahara equation in hydrodynamics), higher order time-stepping schemes can be used and we analyze and implement schemes of order six or less. We derive optimal order error estimates throughout and utilize the schemes to compute the long time dynamics and to characterize the attractors. Various numerical diagnostic tools are implemented, such as the projection of the infinite-dimensional dynamics to one-dimensional return maps that enable us to probe the geometry of the attractors quantitatively. Such results are possible only if computations are carried out for very long times (we provide examples where integrations are carried out for

A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

10^8

Mathematical foundation of the MFS for certain elliptic systems in linear elasticity

time units), and it is shown that the schemes used here are very well suited for such tasks. For illustration, computations are carried out for third order dispersion (the Kawahara equation) as well as fifth order dispersion (the Benney-Lin equation) but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics.