A multi-domain implementation of the pseudo-spectral method and compact finite difference schemes for solving time-dependent differential equations

Abstract : In this dissertation, we introduce new numerical methods for solving time-dependant differential equations. These methods involve dividing the domain of the problem into multiple sub domains. The nonlinearity of the differential equations is dealt with by using a Gauss-Seidel like relaxation or quasilinearisation technique. To solve the linearized iteration schemes obtained we use either higher order compact finite difference schemes or spectral collocation methods and we call the resulting methods the multi-domain compact finite difference relaxation method (MD-CFDRM), multi-domain compact finite difference quasilinearisation method (MD-CFDQLM) and multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) respectively. We test the applicability of these methods in a wide variety of differential equations. The accuracy is compared against other methods as well as other results from literature. The MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. Chaotic and hyperchaotic systems are characterized by high sensitivity to small perturbation on initial data and rapidly changing solutions. Such rapid variations in the solution pose tremendous problems to a number of numerical approximations. We modify the CFDs to be able to deal with such systems of equations. We also used the MD-CFDQLM to solve the nonlinear evolution partial differential equations, namely, the Fisher’s equation, Burgers- Fisher equation, Burgers-Huxley equation and the coupled Burgers’ equations over a large time domain. The main advantage of this approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method for large time domain. We also studied the generalized Kuramoto-Sivashinsky (GKS) equations. The KS equations exhibit chaotic behaviour under certain conditions. We used the multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) to approximate the numerical solutions for the generalized KS equations.M.Sc. (Pure and Applied Mathematics

Multi-level stochastic collocation methods for parabolic and Schrödinger equations

In this thesis, we propose, analyse and implement numerical methods for time-dependent non-linear parabolic and Schrödinger-type equations with uncertain parameters. The discretisation of the parameter space which incorporates the uncertainty of the problem is performed via single- and multi-level collocation strategies. To deal with the possibly large dimension of the parameter space, sparse grid collocation techniques are used to alleviate the curse of dimensionality to a certain extent. We prove that the multi-level method is capable of reducing the overall computational costs significantly. In the parabolic case, the time discretisation is performed via an implicit-explicit splitting strategy of order two which consists shortly speaking of a combination of an implicit trapezoidal rule for the stiff linear part and Heun\u27s method for the non-linear part. In the Schrödinger case, time is discretised via the famous second-order Strang splitting method. For both problem classes we review known error bounds for both discretizations and prove new error bounds for the time discretisations which take the regularity in the parameter space into account. In the parabolic case, a new error bound for the "implicit-explicit trapezoidal method" (IMEXT) method is proved. To our knowledge, this error bound stating second-order convergence of the IMEXT method closes a current gap in the literature. Utilising the aforementioned new error bounds for both problem classes, we can rigorously prove convergence of the single- and multi-level methods. Additionally, cost savings of the multi-level methods compared to the single-level approach are predicted and verifed by numerical examples. The results mentioned above are novel contributions in two areas of mathematics. The first one is (analysis of) numerical methods for uncertainty quantification and the second one is numerical analysis of time-integration schemes for PDEs

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Drift-diffusion models for innovative semiconductor devices and their numerical solution

We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization

The ANOVA decomposition and generalized sparse grid methods for the high-dimensional backward Kolmogorov equation

In this thesis, we discuss numerical methods for the solution of the high-dimensional backward Kolmogorov equation, which arises in the pricing of options on multi-dimensional jump-diffusion processes. First, we apply the ANOVA decomposition and approximate the high-dimensional problem by a sum of lower-dimensional ones, which we then discretize by a θ-scheme and generalized sparse grids in time and space, respectively. We solve the resultant systems of linear equations by iterative methods, which requires both preconditioning and fast matrix-vector multiplication algorithms. We make use of a Linear Program and an algebraic formula to compute optimal diagonal scaling parameters. Furthermore, we employ the OptiCom as non-linear preconditioner. We generalize the unidirectional principle to non-local operators and develop a new matrix-vector multiplication algorithm for the OptiCom. As application we focus on the Kou model. Using a new recurrence formula, the computational complexity of the operator application remains linear in the number of degrees of freedom. The combination of the above-mentioned methods allows us to efficiently approximate the solution of the backward Kolmogorov equation for a ten-dimensional Kou model.Die ANOVA-Zerlegung und verallgemeinerte dünne Gitter für die hochdimensionale Kolmogorov-Rückwärtsgleichung In der vorliegenden Arbeit betrachten wir numerische Verfahren zur Lösung der hochdimensionalen Kolmogorov-Rückwärtsgleichung, die beispielsweise bei der Bewertung von Optionen auf mehrdimensionalen Sprung-Diffusionsprozessen auftritt. Zuerst wenden wir eine ANOVA-Zerlegung an und approximieren das hochdimensionale Problem mit einer Summe von niederdimensionalen Problemen, die wir mit einem θ-Verfahren in der Zeit und mit verallgemeinerten dünnen Gittern im Ort diskretisieren. Wir lösen die entstehenden linearen Gleichungssysteme mit iterativen Verfahren, wofür eine Vorkonditionierung als auch schnelle Matrix-Vektor-Multiplikationsalgorithmen nötig sind. Wir entwickeln ein Lineares Programm und eine algebraische Formel, um optimale Diagonalskalierungen zu finden. Des Weiteren setzen wir die OptiCom als nicht-lineares Vorkonditionierungsverfahren ein. Wir verallgemeinern das unidirektionale Prinzip auf nicht-lokale Operatoren und entwickeln einen für die OptiCom optimierten Matrix-Vektor-Multiplikationsalgorithmus. Als Anwendungsbeispiel betrachten wir das Kou-Modell. Mit einer neuen Rekurrenzformel bleibt die Gesamtkomplexität der Operatoranwendung linear in der Anzahl der Freiheitsgrade. Unter Einbeziehung aller genannten Methoden ist es nun möglich, die Lösung der Kolmogorov-Rückwärtsgleichung für ein zehndimensionales Kou-Modell effizient zu approximieren

Symmetry-based stability theory in fluid mechanics

The present work deals with the stability theory of fluid flows. The central subject is the question under which circumstances a flow becomes unstable. Instabilities are a frequent trigger of laminar-turbulent transitions. Stability theory helps to explain the emergence of structures, e.g. wave-like perturbation patterns. In this context, the use of Lie symmetries allows the classification of existing and the construction of new solutions within the framework of linear stability theory. In addition, a new nonlinear eigenvalue problem (NEVP) is presented, whose derivation is completely based on Lie symmetries. In classical linear stability theory, a normal ansatz is used for perturbations. Another ansatz that has been shown in early work is the Kelvin mode ansatz. In the work of Nold and Oberlack (2013) and Nold et al. (2015) it was shown that these ansätze can be traced back to the Lie symmetries of the linearized perturbation equations. Interestingly, knowledge of the symmetries also allows for the construction of new ansatz functions that go beyond the known ansätze. For a plane rotational shear flow, in addition to the normal mode ansatz, an algebraic mode ansatz with algebraic behavior in time t^s (eigenvalue s) can be constructed. The flow is stable according to Rayleigh's inflection point criterion, which is also confirmed by the algebraic mode ansatz. Furthermore, exact solutions of the eigenfunctions can be found and new stable modes can be determined by asymptotic methods. Thereby, spiral-like structures of the vorticity can be recognized, which propagate in the region with time. Another key result of this work is the formulation and solution of an NEVP based on the Lie symmetries of the Euler equation. It can is shown that an NEVP can be formulated for a class of flows with a constant velocity gradient. These include, for example, linear shear flows, strained flows, and rotating flows. The NEVP for linear shear flows shows a relation to experimental data from turbulent shear flows. It can be theoretically shown that the turbulent kinetic energy scales exponentially with the eigenvalue of the NEVP. The eigenvalue is determined numerically using a parallel spectral solver. Initially, nonlinear terms are neglected. The determined eigenvalues are in the range of known literature values for turbulent shear flows. Furthermore, the NEVPs for plane flows with pure rotation and pure strain are solved. It is shown that the flow is invariant to rotation, while oscillatory eigenfunctions are found in the case of strain. In addition, an algorithm to solve the NEVP including the nonlinear terms is presented. The results allow an exciting insight into a new stability theory and form the basis for further investigation and understanding of the full nonlinear dynamics of the fluid flows based on the NEVP