A Unifying Theory for Runge-Kutta-like Time Integrators: Convergence and Stability

The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. The first major part of this thesis is dedicated to providing a tool for deriving order conditions for NSARK methods. The proposed approach may yield implicit order conditions, which can be rewritten in explicit form using the NB-series of the stages. The obtained explicit order conditions can be further reduced using Gr\"obner bases computations. With the presented approach, it was possible for the first time to obtain conditions for the construction of 3rd and 4th order GeCo as well as 4th order MPRK schemes. Moreover, a new fourth order MPRK method is constructed using our theory and the order of convergence is validated numerically. The second major part is concerned with the stability of nonlinear time integrators preserving at least one linear invariant. We discuss how the given approach generalizes the notion of A-stability. We can prove that investigating the Jacobian of the generating map is sufficient to understand the stability of the nonlinear method in a neighborhood of the steady state. This approach allows for the first time the investigation of several modified Patankar. In the case of MPRK schemes, we compute a general stability function in a way that can be easily adapted to the case of PDRS. Finally, the approach from the theory of dynamical systems is used to derive a necessary condition for avoiding unrealistic oscillations of the numerical approximation.Comment: Doctoral thesi

The Principle of Excluded Middle in Aristotle

Aristoteles zufolge gibt es Prinzipien, welche die ontologischen Konfigurationen des Seien- den sowie deduktives Schließen bestimmen. Das Prinzip vom ausgeschlossenen Dritten (PAD) ist eines dieser Prinzipien. Das PAD erscheint in zwei Versionen: Laut seiner ontologi- schen Version muss alles entweder sein oder nicht sein, und laut seiner ‚psycho-linguisti- schen‘ Version muss alles entweder zu- oder abgesprochen werden. Doch welchen Beitrag leistet das PAD zu Aristoteles’ Philosophie? Ziel dieser Studie ist es, zu zeigen, dass das PAD eine fundamentale Stellung in Aristoteles’ Konzeption von Sein und logischem Denken ein- nimmt: Die ontologische Version des PAD ist eng verknüpft mit Aristoteles’ Kategorienlehre und seinem Begriff des in Aktualität Seienden, während dessen psycho-linguistische Version mit Aristoteles’ Auffassung von Wahrheit und syntaktischer Wohlgeformtheit in Zusammen- hang steht. Bis dato ist jedoch keine umfassende Studie zu Aristoteles’ PAD veröffentlicht worden. Dagegen hat das verwandte Prinzip vom ausgeschlossenen Widerspruch (PAW) weit- aus mehr Aufmerksamkeit erlangt – schließlich hält Aristoteles selbst das PAW für grundle- gender als das PAD. Auch das PAW erscheint in zwei Versionen: Laut seiner ontologischen Version kann nichts zugleich sein und nicht sein, und laut seiner psycho-linguistischen Versi- on kann nichts zugleich zu- und abgesprochen werden. Diese Dissertation bietet die wohl ers- te umfassende Studie zu Aristoteles’ PAD.Aristotle believes there are principles that both determine the ontological configurations of things that are (ὄντα) and govern deductive reasoning. The principle of excluded middle (PEM) is one of them. PEM comes in two versions: its ontological version states that every- thing must either be or not be, and its ‘psycho-linguistic’ version states that everything must either be affirmed or denied. But what exactly does PEM contribute to Aristotle’s philosophy? I aim to show that PEM is fundamental to Aristotle’s views on being and reasoning. Specifi- cally, I argue that there is an intimate connection between PEM’s ontological version and Aristotle’s categorial framework and his conception of being in actuality. PEM’s psycho-lin- guistic version, on the other hand, is intimately connected with Aristotle’s conception of truth and syntactic well-formedness. To date, however, no comprehensive study has been dedicated to Aristotle’s PEM. The cognate principle of non-contradiction (PNC), by contrast, has at- tracted much more scholarly interest. After all, Aristotle himself considers PNC more funda- mental than PEM. PNC too comes in two versions: its ontological version states that nothing can simultaneously be and not be, and its psycho-linguistic version states that nothing can si- multaneously be affirmed and denied. This thesis aims to provide the first comprehensive ac- count of PEM in Aristotle

On the Stability of Modified Patankar Methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity preserving time integrators. In this work, we study the stability properties of the classical modified Patankar--Runge--Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.Comment: 34 pages, 14 Figure

Order conditions for Runge--Kutta-like methods with solution-dependent coefficients

In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent factors. These include the classes of modified Patankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods. Compared to traditional RK methods, the analysis of accuracy and stability of these methods is more complicated. In this work, we provide a comprehensive and unifying theory of order conditions for such RK-like methods, which differ from original RK schemes in that their coefficients are solution-dependent. The resulting order conditions are themselves solution-dependent and obtained using the theory of NB-series, and thus, can easily be read off from labeled N-trees. We present for the first time order conditions for MPRK and GeCo schemes of arbitrary order; For MPRK schemes, the order conditions are given implicitly in terms of the stages. From these results, we recover as particular cases all known order conditions from the literature for first- and second-order GeCo as well as first-, second- and third-order MPRK methods. Additionally, we derive sufficient and necessary conditions in an explicit form for 3rd and 4th order GeCo schemes as well as 4th order MPRK methods. We also present a new 4th order MPRK method within this framework and numerically confirm its convergence rate.Comment: 28 pages, 1 figur

Lyapunov Stability of First and Second Order GeCo and gBBKS Schemes

In this paper we investigate the stability properties of fixed points of the so-called gBBKS and GeCo methods, which belong to the class of non-standard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. The schemes are applied to general linear test equations and proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous first derivatives. As a result, a recently developed stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge--Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.Comment: 31 pages, 7 figure

Determining the Phosphorus Fertilizer Needs for Soils of the High Plains in Oklahoma

Agronom

On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters

Recently, a stability theory has been developed to study the linear stability of modified Patankar--Runge--Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local, but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge--Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global, if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22($\alpha$) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22($\alpha$) schemes with nonnegative Runge-Kutta parameters.Comment: 19 pages, 3 figure

Numerical modeling of the nonlinear mechanical behavior of multilayer geosynthetic system for piggyback landfill expansions

This paper was accepted for publication in the journal Geotextiles and Geomembranes and the definitive published version is available at http://dx.doi.org/10.1016/j.geotexmem.2016.07.004.Numerical modelling techniques have been increasingly used to assess the integrity of engineering works, such as landfills, that involve interactions between multiple geosynthetics GSYs). In piggyback landfill expansions (PBLEs), where a new landfill is built over an older one, such interactions are particularly important because multiple GSYs, natural materials, and waste interact with each over. To obtain reliable numerical results, the real mechanical behavior of the GSYs and of the interfaces between GSYs must be considered. Designers, however, often use simplistic assumptions without further analyzing the implications of these assumptions on the results. Such simplifications mainly concern the nonlinear axial stiffness of GSYs, the strain softening at interfaces between GSYs, and the difference between the compressive and tensile behavior of GSYs. By, considering these key aspects, the present study aims to understand the extent to which the results of numerical calculations can be influenced both by the differing compressive and tensile behavior of GSYs and by the assumption of strain softening at interfaces between GSYs. For this purpose, several numerical models are implemented by using the finite-difference code FLAC 2D on a typical PBLE that involves four GSYs and six interfaces. The present work also applies comprehensive, state-of-the-art numerical modelling to study the interactions between multiple layers of GSYs. This study also investigates the nonlinear axial stiffness of GSYs through a series of uniaxial tensile tests. The numerical results show that, if the GSY axial compressive and tensile characteristics are the same, then tensile force is minimized, which induces significant compressive force in the GSYs. The results also indicate that neglecting strain softening at the interface between GSYs affects interface shear stresses, displacements of GSYs at the interface, and the GSY force distribution, potentially rendering the model unrealistic. Including strain softening, however, allows the assessment (location) of unstable areas along the interface where large displacements occur

Hur har digitaliseringen påverkat revisorns arbetssätt och roll? : En kvalitativ studie om digitaliseringen inom revisionsbranschen

Digitization has changed the auditing profession in different ways. Audits can now be carried out digitally and audit firms have switched from the analogue to the digital way of working. This is because the audit process has been digitized and many work steps that have previously been time-consuming to carry out manually today can be carried out efficiently digitally. This study aims to investigate how digitalisation has changed the auditor's working methods and function. What will be examined is which working methods have changed, which skills it requires from the auditor and how the auditor's function is affected. The study is characterized by a qualitative research approach where six semi-structured interviews were conducted with three different audit firms. These interviews together with the study's theoretical frame of reference lay the foundation for the study's results. It has been established by previous research that working methods in the audit process have changed in step with digitalisation. Therefore, the authors have further researched this through data to confirm the previous research. The results show that the audit process has changed and facilitated the auditor's tasks. With the help of the digital tools and systems, the auditor can now carry out his work digitally and easily search for the financial items needed, as well as through cloud technology manage data and communicate with the client. These changes free up time for the auditor to create added value for the client through advisory services

Hur har digitaliseringen påverkat revisorns arbetssätt och roll? : En kvalitativ studie om digitaliseringen inom revisionsbranschen

Digitization has changed the auditing profession in different ways. Audits can now be carried out digitally and audit firms have switched from the analogue to the digital way of working. This is because the audit process has been digitized and many work steps that have previously been time-consuming to carry out manually today can be carried out efficiently digitally. This study aims to investigate how digitalisation has changed the auditor's working methods and function. What will be examined is which working methods have changed, which skills it requires from the auditor and how the auditor's function is affected. The study is characterized by a qualitative research approach where six semi-structured interviews were conducted with three different audit firms. These interviews together with the study's theoretical frame of reference lay the foundation for the study's results. It has been established by previous research that working methods in the audit process have changed in step with digitalisation. Therefore, the authors have further researched this through data to confirm the previous research. The results show that the audit process has changed and facilitated the auditor's tasks. With the help of the digital tools and systems, the auditor can now carry out his work digitally and easily search for the financial items needed, as well as through cloud technology manage data and communicate with the client. These changes free up time for the auditor to create added value for the client through advisory services