4,578 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
Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
The symmetry algebra of the real elliptic Liouville equation is an
infinite-dimensional loop algebra with the simple Lie algebra as its
maximal finite-dimensional subalgebra. The entire algebra generates the
conformal group of the Euclidean plane . This infinite-dimensional algebra
distinguishes the elliptic Liouville equation from the hyperbolic one with its
symmetry algebra that is the direct sum of two Virasoro algebras. Following a
discretisation procedure developed earlier, we present a difference scheme that
is invariant under the group and has the elliptic Liouville equation
in polar coordinates as its continuous limit. The lattice is a solution of an
equation invariant under and is itself invariant under a subgroup of
, namely the rotations of the Euclidean plane
An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry.\ud
We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities
Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
The main purpose of this article is to show how symmetry structures in
partial differential equations can be preserved in a discrete world and
reflected in difference schemes. Three different structure preserving
discretizations of the Liouville equation are presented and then used to solve
specific boundary value problems. The results are compared with exact solutions
satisfying the same boundary conditions. All three discretizations are on four
point lattices. One preserves linearizability of the equation, another the
infinite-dimensional symmetry group as higher symmetries, the third one
preserves the maximal finite-dimensional subgroup of the symmetry group as
point symmetries. A 9-point invariant scheme that gives a better approximation
of the equation, but significantly worse numerical results for solutions is
presented and discussed
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Algorithms for Mappings and Symmetries of Differential Equations
Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points
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