On Compact Finite Difference Schemes With Applications To Moving Boundary Problems

Compact finite differences are introduced with the purpose of developing compact methods of higher order for the numerical solution of ordinary and elliptic partial differential equations.;The notion of poisedness of a compact finite difference is introduced. It is shown that if the incidence matrix of the underlying interpolation problem contains no odd unsupported sequences then the Polya conditions are necessary and sufficient for poisedness.;A Pade Operator method is used to construct compact formulae valid for uniform three point grids. A second Function-Theoretic method extends compact formulae to variably-spaced three point grids with no deterioration in the order of the truncation error.;A new fourth order compact method (CI4) leading to matrix systems with block tridiagonal structure, is applied to boundary value problems associated with second order ordinary differential equations. Numerical experiments with both linear and nonlinear problems and on uniform and nonuniform grids indicate rates of convergence of four.;An application is considered to the time-dependent one-dimensional nonlinear Burgers\u27 equation in which an initial sinusoidal disturbance develops a very sharp boundary layer. It is found that the CI4 method, with a small number of points placed on a highly stretched grid, is capable of accurately resolving the boundary layer.;A new method (LCM) based on local polynomial collocation and Gauss-type quadrature and leading to matrix systems with block tridiagonal structure, is used to generate high order compact methods for ordinary differential equations. A tenth order method is shown to be considerably more efficient than the CI4 method.;A new fourth order compact method, based on the CI4 method, is developed for the solution, on variable grids, of two-dimensional, time independent elliptic partial differential equations. The method is applied to the ill-posed problem of calculating the interface in receding Hele-Shaw flow. Comparisons with exact solutions indicate that the numerical method behaves as expected for early times.;Finally, in an application to the simulation of contaminant transport within a porous medium under an evolving free surface, new fourth order explicit compact expressions for mixed derivatives are developed

A new code for equilibriums and quasiequilibrium initial data of compact objects

We present a new code, named COCAL - Compact Object CALculator, for the computation of equilibriums and quasiequilibrium initial data sets of single or binary compact objects of all kinds. In the cocal code, those solutions are calculated on one or multiple spherical coordinate patches covering the initial hypersurface up to the asymptotic region. The numerical method used to solve field equations written in elliptic form is an adaptation of self-consistent field iterations in which Green's integral formula is computed using multipole expansions and standard finite difference schemes. We extended the method so that it can be used on a computational domain with excised regions for a black hole and a binary companion. Green's functions are constructed for various types of boundary conditions imposed at the surface of the excised regions for black holes. The numerical methods used in cocal are chosen to make the code simpler than any other recent initial data codes, accepting the second order accuracy for the finite difference schemes. We perform convergence tests for time symmetric single black hole data on a single coordinate patch, and binary black hole data on multiple patches. Then, we apply the code to obtain spatially conformally flat binary black hole initial data using boundary conditions including the one based on the existence of equilibrium apparent horizons.Comment: Revised version with a new title, 29 page

Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions

We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampere equation using finite dimensional spaces of piecewise polynomial C0 or C1 functions. Standard discretizations of the type considered in this paper have been previous analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Amp\`ere equation