Simulation of long-term influence from technical systems on permafrost with various short-scale and hourly operation modes in Arctic region

Technogenic and climatic influences have a significant impact on the degradation of permafrost. Long-term forecasts of such changes during long-time periods have to be taken into account in the oil and gas and construction industries in view to development the Arctic and Subarctic regions. There are considered constantly operating technical systems (for example, oil and gas wells) that affect changes in permafrost, as well as the technical systems that have a short-term impact on permafrost (for example, flare systems for emergency flaring of associated gas). The second type of technical systems is rather complex for simulation, since it is required to reserve both short and long-scales in computations with variable time steps describing the complex technological processes. The main attention is paid to the simulation of long-term influence on the permafrost from the second type of the technical systems. © 2017 Author(s).The work was supported by Russian Foundation for Basic Research 16–01–00401 and program of scientific research UrB RAS 15–16–1–10

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed. © 2017 Author(s).The work was supported by Russian Foundation for Basic RTesearch 16–01–00401

Uniqueness of iterative positive solutions for the singular infinite-point p-Laplacian fractional differential system via sequential technique

By sequential techniques and mixed monotone operator, the uniqueness of positive solution for singular p-Laplacian fractional differential system with infinite-point boundary conditions is obtained. Green's function is derived, and some useful properties of Green' function are obtained. Based on these new properties, the existence of unique positive solutions is established, moreover, an iterative sequence and a convergence rate are given, which are important for practical application, and an example is given to demonstrate the validity of our main results

A Computational model for fluid-porous structure interaction

This work utilizes numerical models to investigate the importance of poroelasticity in Fluid- Structure Interaction, and to establish a connection between the apparent viscoelastic behavior of the structure part and the intramural filtration flow. We discuss a loosely coupled computational framework for modeling multiphysics systems of coupled flow and mechanics via finite element method. Fluid is modeled as an incompressible, viscous, Newtonian fluid using the Navier-Stokes equations and the structure domain consists of a thick poroelastic material, which is modeled by the Biot system. Physically meaningful interface conditions are imposed on the discrete level via mortar finite elements or Nitsche's coupling. We further discuss the use of our loosely coupled non-iterative time-split formulation as a preconditioner for the monolithic scheme. We further investigate the interaction of an incompressible fluid with a poroelastic structure featuring possibly large deformations, where the assumption of large deformations is taken into account by including the full strain tensor. We use this model to study the influence of different parameters on energy dissipation in a poroelastic medium. The numerical results investigate the effects of poroelastic parameters on the pressure wave propagation, filtration of the incompressible fluid through the porous media, and the structure displacement

Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand

Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed

Numerical Integrators for Maxwell-Klein-Gordon and Maxwell-Dirac Systems in Highly to Slowly Oscillatory Regimes

Maxwell-Klein-Gordon (MKG) and Maxwell-Dirac (MD) systems physically describe the mutual interaction of moving relativistic particles with their self-generated electromagnetic field. Solving these systems in the nonrelativistic limit regime, i.e. when the speed of light $c$ formally tends to infinity, is numerically very delicate as the solution becomes highly oscillatory in time. In order to resolve the oscillations, standard time integrations schemes require severe restrictions on the time step $\tau\sim c^{-2}$ depending on the small parameter $c^{-2}$ which leads to high computational costs. Within this thesis we propose and analyse two types of numerical integrators to efficiently integrate the MKG and MD systems in highly oscillatory nonrelativistic limit regimes to slowly oscillatory relativistic regimes. The idea for the first type relies on asymptotically expanding the exact solution in the small parameter $c^{-1}$. This results in non-oscillatory Schrödinger-Poisson (SP) limit systems which can be solved efficiently by using classical splitting schemes. We will see that standard Strang splitting schemes, applied to the latter SP systems with step size $\tau$, allow error bounds of order $\mathcal{O}(\tau^2+c^{-N})$ for $N\in \mathbb N$ without any time step restriction. Thus, in the nonrelativistic limit regime $c\rightarrow\infty$ these methods are very efficient and allow an accurate approximation to the exact solution. The second type of numerical integrator is based on "twisted variables" which have been originally introduced for the Klein-Gordon equation in [Baumstark/Faou/Schratz, 2017]. In the case of MKG and MD systems however, due to the strong nonlinear coupling between the components of the solution, the construction and analysis is much more involved. We thereby exploit the main advantage of the "twisted variables" that they have bounded derivatives with respect to $c\rightarrow\infty$. Together with a splitting approach, this allows us to construct an exponential-type splitting method which is first order accurate in time uniformly in $c$. Due to error bounds of order $\mathcal{O}(\tau)$ independent of $c$ without any restriction on the time step $\tau$, these schemes are efficient in highly to slowly oscillatory regimes