Grain transport and rural credit in Mozambique: solving the space-time problem

The Boundary Element Method (BEM) allows efficient solution of partial differential equations whose kernel functions are known. The heat equation is one of these candidates when the thermal parameters are assumed constant (linear model). When the model involves large physical domains and time simulation intervals the amount of information that must be stored increases significantly. This drawback can be circumvented by using advanced strategies, as for example the multi-poles technique. We propose radically different approach that leads to a separated solution of the space and time problems within a non-incremental integration strategy. The technique is based on the use of a space-time separated representation of the unknown field that, introduced in the residual weighting formulation, allows to define a separated solution of the resulting weak form. The spatial step can be then treated by invoking the standard BEM for solving the resulting steady state problem defined in the physical space. Then, the time problem that results in an ordinary first order differential equation is solved using any standard appropriate integration technique (e.g. backward finite differences). When considering the nonlinear heat equation, the BEM cannot be easily applied because its Green’s kernel is generally not known but the use of the PGD presents the advantage of rewriting the problem in such a way that the kernel is now clearly known. Indeed, the system obtained by the PGD is composed of a Poisson equation in space coupled with an ODE in time so that the use of the BEM for solving the spatial part of the problem is efficient. During the solving, we must however separate the nonlinear term into a space-time representation that can limit the method in terms of CPU time and storage, that is why we introduce in the second part of the paper a new approach combining the PGD and the Asymptotic Numerical Method (ANM) in order to efficiently treat the nonlinearity.

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Mathematics and the Space-Time Problem

Mathematics Magazine ◽

10.2307/3029447 ◽

1952 ◽

Vol 25 (3) ◽

pp. 147

Author(s):

Roger Osborn

Keyword(s):

Space Time ◽

Time Problem

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A space-time certified reduced basis method for quasilinear parabolic partial differential equations

Advances in Computational Mathematics ◽

10.1007/s10444-021-09860-z ◽

2021 ◽

Vol 47 (3) ◽

Author(s):

Michael Hinze ◽

Denis Korolev

Keyword(s):

Approximation Error ◽

Posteriori Error ◽

Space Time ◽

A Posteriori ◽

Reduced Basis ◽

Element Discretization ◽

A Posteriori Error ◽

Time Problem ◽

Time Marching ◽

Computable Bound

AbstractIn this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

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Reactor Depletion Analyzed as a Space-Time Problem

Nuclear Science and Engineering ◽

10.13182/nse72-a22567 ◽

1972 ◽

Vol 49 (4) ◽

pp. 482-488

Author(s):

William J. Westlake ◽

A. F. Henry

Keyword(s):

Space Time ◽

Time Problem

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GRAVITATIONAL THEORY WITH A DYNAMICAL SPACE–TIME

International Journal of Modern Physics A ◽

10.1142/s0217751x10050317 ◽

2010 ◽

Vol 25 (21) ◽

pp. 4081-4099 ◽

Cited By ~ 14

Author(s):

E. I. GUENDELMAN

Keyword(s):

Vector Field ◽

Inertial Frame ◽

Gravitational Theory ◽

Momentum Tensor ◽

Space Time ◽

Time Problem ◽

Matter Model ◽

Two Measures ◽

Tensor Formula ◽

Flat Coordinates

A gravitational theory involving a vector field χμ, which has the properties of a dynamical space–time, is studied. The variation of the action with respect to χμ gives the covariant conservation of an energy–momentum tensor [Formula: see text]. Studying the theory in a background that has Killing vectors and Killing tensors, we find appropriate shift symmetries of the field χμ which lead to conservation laws. The energy–momentum that is the source of gravity [Formula: see text] is different but related to [Formula: see text] and the covariant conservation of [Formula: see text] determines in general the vector field χμ. When [Formula: see text] is chosen to be proportional to the metric, the theory coincides with the Two Measures Theory, which has been studied before in relation to the Cosmological Constant Problem. When the matter model consists of point particles, or strings, the form of [Formula: see text], solutions for χμ are found. In a locally inertial frame, the vector field corresponds to the locally flat coordinates. For the case of a string gas cosmology, we find that the Milne universe can be a solution, where the gas of strings does not curve the space–time since although [Formula: see text], [Formula: see text], as a model for the early universe, this solution is also free of the horizon problem. There may also be an application to the "time problem" of quantum cosmology.

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