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Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations
Olivier Sarbach,Manuel Tiglio
Living Reviews in Relativity , 2012,
Abstract: Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.
Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations  [PDF]
Gino Biondini,Guenbo Hwang
Physics , 2008, DOI: 10.1088/0266-5611/24/6/065011
Abstract: We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary value problems for the discrete analogue of both the linear and the nonlinear Schrodinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write down the soliton solutions.
Fully discrete hyperbolic initial boundary value problems with nonzero initial data  [PDF]
Jean-Fran?ois Coulombel
Mathematics , 2014,
Abstract: The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in ? 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b].
Construction of upper and lower solutions for singular discrete initial and boundary value problems via inequality theory  [cached]
Lü Haishen,O'Regan Donal
Advances in Difference Equations , 2005,
Abstract: We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.
Construction of upper and lower solutions for singular discrete initial and boundary value problems via inequality theory
Haishen Lü,Donal O'Regan
Advances in Difference Equations , 2005, DOI: 10.1155/ade.2005.205
Abstract: We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.
Integrable boundary conditions and modified Lax equations  [PDF]
Jean Avan,Anastasia Doikou
Physics , 2007, DOI: 10.1016/j.nuclphysb.2008.04.004
Abstract: We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding transfer matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix.
Integrable initial boundary value problems  [PDF]
I. T. Habibullin
Physics , 2002,
Abstract: The problem of searching boundary value problems for soliton equations consistent with the integrability property is discussed. A method of describing integrals of motion for the integrable initial boundary value problems for the Kadomtsev-Petviashvili equation is suggested via Green identity.
From discrete particles to continuum fields near a boundary  [PDF]
Thomas Weinhart,Anthony R. Thornton,Stefan Luding,Onno Bokhove
Physics , 2011,
Abstract: An expression for the stress tensor near an external boundary of a discrete mechanical system is derived explicitly in terms of the constituents' degrees of freedom and interaction forces. Starting point is the exact and general coarse graining formulation presented by Goldhirsch in [I. Goldhirsch, Gran. Mat., 12(3):239-252, 2010], which is consistent with the continuum equations everywhere but does not account for boundaries. Our extension accounts for the boundary interaction forces in a self-consistent way and thus allows the construction of continuous stress fields that obey the macroscopic conservation laws even within one coarse-graining width of the boundary. The resolution and shape of the coarse-graining function used in the formulation can be chosen freely, such that both microscopic and macroscopic effects can be studied. The method does not require temporal averaging and thus can be used to investigate time-dependent flows as well as static and steady situations. Finally, the fore-mentioned continuous field can be used to define 'fuzzy' (highly rough) boundaries. Two discrete particle method simulations are presented in which the novel boundary treatment is exemplified, including a chute flow over a base with roughness greater than a particle diameter.
Initial-Boundary Value Problems for Parabolic Equations  [PDF]
Magnus Fontes
Mathematics , 2008,
Abstract: We prove new existence and uniqueness results for weak solutions to non-homogeneous initial-boundary value problems for parabolic equations modeled on the evolution of the p-Laplacian.
Discrete Fourier multipliers and cylindrical boundary value problems  [PDF]
Tobias Nau,Robert Denk
Mathematics , 2011,
Abstract: We consider operator-valued boundary value problems in $(0,2\pi)^n$ with periodic or, more generally, $\nu$-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains $(0,2\pi)^n\times V$ with $\nu$-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal $L^q$-regularity for such problems.
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