The Computational Singular Perturbation (CSP) method, developed by Lam and Goussis [Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 931–941], is a commonly-used method for finding approximations of slow manifolds in systems of ordinary differential equations (ODEs) with multiple time scales. The validity of the CSP method was established fo...

In this work we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the Fokker–Planck equation and some similar equations. ...

In this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting, alternating direction or, more generally, fractional step methods. Such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive Runge–Kutta method, which is called the fractional step Runge–Kutta met...

Freak waves are extreme ocean waves that are not predicted by traditional wave probability distributions. We study wave energy focusing, as a possible mechanism for freak wave formation, in a three-dimensional (3D) Numerical Wave Tank (NWT) solving fully nonlinear potential ow equations. The NWT combines a higher-order 3D-BEM and a Mixed-Eulerian-Lagrangian time updating of the free surface, ba...

In this paper we present a new unstructured spectral element method for use on triangular and tetrahedral subdomains. The algorithm maintains the accuracy and efficiency of standard quadrilateral/hexahedral spectral element methods but offers greater adaptivity. Standard unstructured meshes can be used and the order of the polynomial expansion can be varied within each triangular/tetrahedral su...

We present a high-order spectral element method for solving layered media scattering problems featuring an operator that transparently enforces the far-field boundary condition. The incorporation of this Dirichlet-to-Neumann (DtN) map into the spectral element framework is a novel aspect of this work, and the resulting method can accommodate plane-wave radiation of arbitrary angle of incidence....

Title of dissertation: FAST SOLVERS FOR MODELS OF FLUID FLOW WITH SPECTRAL ELEMENTS P. Aaron Lott, Doctor of Philosophy, 2008 Dissertation directed by: Professor Howard Elman Department of Computer Science Institute for Advanced Computer Studies & Associate Research Professor Anil Deane Institute for Physical Sciences and Technology We introduce a preconditioning technique based on Domain Decom...

A spectral element method is described for the flow of a viscoelastic fluid past a sphere in a tube. An approximate Jacobian is used to solve the system of nonlinear algebraic equations derived from the spectral element discretization as a cost-effective alternative to the full Newton method. The linearized equations are solved using a preconditioned generalised minimal residual (GMRES) method....

An unstructured nodal spectral-element method for the Navier-Stokes equations is developed in this paper. The method is based on a triangular and tetrahedral rational approximation and an easy-to-implement nodal basis which fully enjoys the tensorial product property. It allows arbitrary triangular and tetrahedral mesh, affording greater flexibility in handling complex domains while maintaining...

A very simple and eÆcient nite element method is introduced for two and three dimensional viscous incompressible ows using the vorticity formulation. The method relies on recasting traditional nite element method in the spirit of the high order accurate nite di erence methods introduced by the authors in [10]. Optimal accuracy of arbitrary order can be achieved using standard nite element or sp...