80,499 research outputs found
A class of well-posed approximations for constrained second order hyperbolic equations
The purpose of this paper is to present a new family of numerical methods for the approximation of second order hyperbolic partial differential equations submitted to a convex constraint on the solution. The principle is a singular modification of the mass matrix obtained by the mean of different discretizations of the solution and of its time derivative. The major interest of these methods is that the semi-discretized problem is well-posed and energy conserving. Numerical experiments show that this is a crucial property to build stable numerical schemes
Superconvergence in Iterated Solutions of Integral Equations
In this thesis, we investigate the superconvergence phenomenon of the iterated numerical solutions for the Fredholm integral equations of the second kind as well as a class of nonliner Hammerstein equations. The term superconvergence was first described in the early 70s in connection with the solution of two-point boundary value problems and other related partial differential equations. Superconvergence in this context was understood to mean that the order of convergence of the numerical solutions arising from the Galerkin as well as the collocation method is higher at the knots than we might expect from the numerical solutions that are obtained by applying a class of piecewise polynomials as approximating functions. The type of superconvergence that we investigate in this thesis is different. We are interested in finding out whether or not we obtain an enhancement in the global rate of convergence when the numerical solutions are iterated through integral operators. A general operator approximation scheme for the second kind linear equation is described that can be used to explain some of the existing superconvergence results. Moreover, a corollary to the general approximation scheme will be given which can be used to establish the superconvergence of the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for Hammerstein and weakly singular Hammerstein equations and its corresponding superconvergence phenomena for the iterated solutions. The type of regularities that the solution of weakly singular Hammerstein equations possess is investigated. Subsequently, we establish the singularity preserving Galerkin method for Hammerstein equations. Finally, the superconvergence results for the iterated solutions corresponding to this method will be described
Second order parameter-uniform numerical method for a partially singularly perturbed linear system of reaction-diusion type
A partially singularly perturbed linear system of second order ordinary
differential equations of reaction-diffusion type with given
boundary conditions is considered. The leading terms of first equations are multiplied by small positive singular
perturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first components
of the solution exhibit overlapping layers and the remaining components have less-severe overlapping layers. Shishkin
piecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to all
the parameters. Numerical illustrations are presented in support of the theory
Principled interpolation of Green's functions learned from data
We present a data-driven approach to mathematically model physical systems
whose governing partial differential equations are unknown, by learning their
associated Green's function. The subject systems are observed by collecting
input-output pairs of system responses under excitations drawn from a Gaussian
process. Two methods are proposed to learn the Green's function. In the first
method, we use the proper orthogonal decomposition (POD) modes of the system as
a surrogate for the eigenvectors of the Green's function, and subsequently fit
the eigenvalues, using data. In the second, we employ a generalization of the
randomized singular value decomposition (SVD) to operators, in order to
construct a low-rank approximation to the Green's function. Then, we propose a
manifold interpolation scheme, for use in an offline-online setting, where
offline excitation-response data, taken at specific model parameter instances,
are compressed into empirical eigenmodes. These eigenmodes are subsequently
used within a manifold interpolation scheme, to uncover other suitable
eigenmodes at unseen model parameters. The approximation and interpolation
numerical techniques are demonstrated on several examples in one and two
dimensions
- …