Special Issue "Symmetry in Splitting Methods for Partial and Stochastics Differential Equations: Theory and Applications"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry".

Deadline for manuscript submissions: 31 October 2020.

Special Issue Editor

PD Dr. Jürgen Geiser
Website SciProfiles
Guest Editor
The Institute of Theoretical Electrical Engineering, Ruhr University of Bochum, Universitätsstrasse 150, D-44801 Bochum, Germany
Interests: numerical analysis; computational sciences; multiscale analysis; partial differential equations; stochastic differential equations; coupling and decomposition methods; interface analysis; analysis of parallel methods; asynchronous methods; mathematical physics; statistical mechanics; kinetic theory

Special Issue Information

Dear Colleagues,

Splitting methods have been applied to partial and stochastic differential equations for many years and provide the advantage of decomposing differential equations into simpler solvable sub-differential equations. Optimization and acceleration of such splitting methods can be achieved by applying symmetries in the underlying splitting ideas, e.g., by decomposing into symmetrical sub-equation parts (symmetrical splitting or Strang-splitting methods), symmetrical upper and lower diagonal matrix-operators (Waveform relaxation methods), and operators with symmetries in the multidimensional space matrices (ADI, LOD, and dimension splitting).

In general, we consider symmetrical structures from the beginning in the modelling equations, while we decompose them into similar processes, which are important in transport and flow problems. In the following stage, when we apply discretization methods in space, we consider linear and nonlinear systems of  differential equations with matrix operators, so that we can decompose such operators in simpler and symmetrical parts. Finally, in the last stage, we apply parallel methods to accelerate the solver processes, e.g., balancing or symmetries in the distributions. Such a careful study allows to improve the solver process for numerical simulations and to achieve higher order, more efficient, and more scalable methods.

This Special Issue is addressed to scientific researchers working in the field of applied mathematics and, more specifically, to mathematicians, physicists, engineers, chemists, computer scientists. The main aim of this Special Issue is to provide a platform for the discussion of the major research challenges and achievements related to splitting methods in time and space with respect to their symmetries. Research articles as well as review articles are welcome.

The topics include, but are not limited, to:

  • Theoretical splitting analysis based on symmetry
  • Splitting methods related to multiscale and multicomponent models
  • Applications and simulation of transport, flow, and dynamical systems with symmetries in the models
  • Parallel methods with symmetry ideas
  • Theory and application of numerical methods in transport and flow models solved with splitting approaches

Prof. Juergen Geiser
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Symmetric Splitting Methods
  • Operator Splitting Methods
  • Space or time decomposition Methods
  • Multidimensional splitting methods
  • Transport and Flow problems
  • Iterative splitting Methods
  • Parallel Methods with symmetrical ideas
  • Symmetries in spectral splitting methods
  • Symmetrical Modelling

Published Papers (2 papers)

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Research

Open AccessArticle
Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions
Symmetry 2020, 12(8), 1265; https://doi.org/10.3390/sym12081265 - 01 Aug 2020
Abstract
We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function [...] Read more.
We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence. Full article
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Open AccessArticle
Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations
Symmetry 2020, 12(3), 383; https://doi.org/10.3390/sym12030383 - 03 Mar 2020
Cited by 1
Abstract
In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic [...] Read more.
In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods. Full article
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