975 research outputs found
An almost symmetric Strang splitting scheme for nonlinear evolution equations
In this paper we consider splitting methods for the time integration of
parabolic and certain classes of hyperbolic partial differential equations,
where one partial flow can not be computed exactly. Instead, we use a numerical
approximation based on the linearization of the vector field. This is of
interest in applications as it allows us to apply splitting methods to a wider
class of problems from the sciences.
However, in the situation described the classic Strang splitting scheme,
while still a method of second order, is not longer symmetric. This, in turn,
implies that the construction of higher order methods by composition is limited
to order three only. To remedy this situation, based on previous work in the
context of ordinary differential equations, we construct a class of Strang
splitting schemes that are symmetric up to a desired order.
We show rigorously that, under suitable assumptions on the nonlinearity,
these methods are of second order and can then be used to construct higher
order methods by composition. In addition, we illustrate the theoretical
results by conducting numerical experiments for the Brusselator system and the
KdV equation
Reduction operators and exact solutions of generalized Burgers equations
Reduction operators of generalized Burgers equations are studied. A
connection between these equations and potential fast diffusion equations with
power nonlinearity -1 via reduction operators is established. Exact solutions
of generalized Burgers equations are constructed using this connection and
known solutions of the constant-coefficient potential fast diffusion equation.Comment: 7 page
Parameter estimation for semilinear SPDEs from local measurements
This work contributes to the limited literature on estimating the diffusivity
or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that
the solution is measured locally in space and over a finite time interval, we
show that the augmented maximum likelihood estimator introduced in Altmeyer,
Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs
that satisfy some abstract, and verifiable, conditions. The proofs of
asymptotic results are based on splitting the solution in linear and nonlinear
parts and fine regularity properties in -spaces. The obtained general
results are applied to particular classes of equations, including stochastic
reaction-diffusion equations. The stochastic Burgers equation, as an example
with first order nonlinearity, is an interesting borderline case of the general
results, and is treated by a Wiener chaos expansion. We conclude with numerical
examples that validate the theoretical results.Comment: corrected versio
Operator splitting for the Benjamin-Ono equation
In this paper we analyze operator splitting for the Benjamin-Ono equation,
u_t = uu_x + Hu_xx, where H denotes the Hilbert transform. If the initial data
are sufficiently regular, we show the convergence of both Godunov and Strang
splitting.Comment: 18 Page
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
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