62,111 research outputs found
Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations
In this paper we present an algorithmic approach to the generation of fully
conservative difference schemes for linear partial differential equations. The
approach is based on enlargement of the equations in their integral
conservation law form by extra integral relations between unknown functions and
their derivatives, and on discretization of the obtained system. The structure
of the discrete system depends on numerical approximation methods for the
integrals occurring in the enlarged system. As a result of the discretization,
a system of linear polynomial difference equations is derived for the unknown
functions and their partial derivatives. A difference scheme is constructed by
elimination of all the partial derivatives. The elimination can be achieved by
selecting a proper elimination ranking and by computing a Gr\"obner basis of
the linear difference ideal generated by the polynomials in the discrete
system. For these purposes we use the difference form of Janet-like Gr\"obner
bases and their implementation in Maple. As illustration of the described
methods and algorithms, we construct a number of difference schemes for Burgers
and Falkowich-Karman equations and discuss their numerical properties.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Consistency Conditions for Fundamentally Discrete Theories
The dynamics of physical theories is usually described by differential
equations. Difference equations then appear mainly as an approximation which
can be used for a numerical analysis. As such, they have to fulfill certain
conditions to ensure that the numerical solutions can reliably be used as
approximations to solutions of the differential equation. There are, however,
also systems where a difference equation is deemed to be fundamental, mainly in
the context of quantum gravity. Since difference equations in general are
harder to solve analytically than differential equations, it can be helpful to
introduce an approximating differential equation as a continuum approximation.
In this paper implications of this change in view point are analyzed to derive
the conditions that the difference equation should satisfy. The difference
equation in such a situation cannot be chosen freely but must be derived from a
fundamental theory. Thus, the conditions for a discrete formulation can be
translated into conditions for acceptable quantizations. In the main example,
loop quantum cosmology, we show that the conditions are restrictive and serve
as a selection criterion among possible quantization choices.Comment: 33 page
Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients
In this paper hyperbolic partial differential equations with random
coefficients are discussed. Such random partial differential equations appear
for instance in traffic flow problems as well as in many physical processes in
random media. Two types of models are presented: The first has a time-dependent
coefficient modeled by the Ornstein--Uhlenbeck process. The second has a random
field coefficient with a given covariance in space. For the former a formula
for the exact solution in terms of moments is derived. In both cases stable
numerical schemes are introduced to solve these random partial differential
equations. Simulation results including convergence studies conclude the
theoretical findings
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