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

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

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