This dissertation is a summary of the graduate study in the past few years. In first part, we develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the nonlocal fractional Laplacian operator can be trivially evaluated, leading to very efficient algorithms for PDEs involving spectral fractional Laplacian. We provide a rigorous error analysis for the proposed methods, as well as ample numerical results to show their effectiveness.<div><br>In second part, we propose a method suitable for the computation of quasiperiodic interface, and apply it to simulate the interface between ordered phases in Lifschitz-Petrich model, which can be quasiperiodic. The function space, initial and boundary conditions are carefully chosen such that it fix the relative orientation and displacement, and we follow a gradient flow to let the interface and its optimal structure. The gradient flow is discretized by the scalar auxiliary variable (SAV) approach in time, and spectral method in space using quasiperiodic Fourier series and generalized Jacobi<br>polynomials. We use the method to study interface between striped, hexagonal and dodecagonal phases, especially when the interface is quasiperiodic. The numerical examples show that our method is efficient and accurate to successfully capture the interfacial structure.</div>

Heat transfer study on gas turbine blades is very important due to the resultant increase in cycle thermal efficiency. This study is focused on the heat transfer effects on a reference nozzle guide vane and test rig component preparation in heat and power technology division at KTH. In order to prepare the current test rig for heat transfer experiments, some feature should be changed in the current layout to give a nearly instant temperature rise for heat transfer measurement. The heater mesh component is the main component to be added to the current test rig. Some preliminary design parameters were set and the necessary power for the heater mesh to achieve required step temperature rise was calculated. For the next step, it is needed to estimate the heat transfer coefficient and the other parameters for study on the reference blade using numerical methods. Boundary layer analysis is very important in heat transfer modeling so to model the reference blade heat transfer and boundary layer properties, a 2D boundary layer code TEXSTAN is used and the velocity distribution around the vane was set to an input dataset file. After elements refinement to ensure the numerical accuracy of TEXSTAN code, various turbulence modeling was check to predict the heat transfer coefficient and boundary layer assessments. It was concluded from TEXTAN calculations that both suction and pressure side have transition flow while for the suction side it was predicted that the flow regime at trailing edge is fully turbulent. Based on the Abu-Ghannam –Shaw Transition model and by the aid of shape factor data, momentum Reynolds number and various boundary layer properties, it was concluded that the pressure side remains in transient region.

>Magister Scientiae - MSc / In this thesis, we consider some interesting mathematical models arising in ecology. Our focus is on the exploration of robust numerical solvers which are applicable to models arising in mathematical ecology. To begin with, we consider a simple but nonlinear second-order time-dependent partial differential equation, namely, the Allen-Cahn equation. We discuss the construction of a class of exponential time differencing methods to solve this particular problem. This is then followed by a discussion on the extension of this approach to solve a more difficult fourth-order time-dependent partial differential equation, namely, Kuramoto-Sivashinsky equation. This equation is nonlinear. Further applications include the extension of this approach to solve a complex predator-prey system which is a system of fourth-order time-dependent non-linear partial differential equations. For each of these differential equation models, we presented numerical simulation results. / 2025

The aim of this thesis was to simulate a part of a proposed experiment by Lev Vaidman by using Bohmian mechanics. To do this a numerical method for solving the Schrödinger equation and theguiding equation was created, with several ways of making the simulation more efficient.To make the simulation work more efficiently the Schrödinger equation was applied to only a small region of the whole setup. This region followed the wavefunction of significant values and could change size during the simulation. A beam splitter was constructed in the form of a thin potential barrier. The beam splitter was tested to verify that the reflected and transmitted angles agreed with expectations. A virtual detector was constructed and used for the calibration of the beam splitter to determine which potential resulted in dividing the wave packet into two wave packets of equal intensity. A fixed angle mirror was used for testing the reflection of a wave packet for the reflected angle and concluded that it agreed with the expectations for it. Testing a time dependent mirror for different frequencies and amplitudes was performed, with the result that the numerical method could be used to determine the particles’ trajectories. These results were used to construct a larger setup that was a small part of Vaidman’s proposed experiment. These setups were done in several version. All setups had one wave packet that went through one beam splitter and separated into two wave packets. These two wave packets reflected at two mirrors with different frequencies and then interfered with each other at either free space or at another beam splitter. The result of the simulation of these setups was that the particles’ trajectories could be calculated with the guiding equation. / Syftet med denna avhandling var att simulera en del av det föreslagna experimentet av Lev Vaidman med hjälp av Bohmsk mekanik. För att göra detta skapades en numerisk metod för att lösa Schrödingerekvationen och den ledande ekvationen, ”the guiding equation”, med flera sätt att effektivisera simuleringen. För att effektivisera simuleringen tillämpades Schrödingerekvationen på endast en liten region i hela uppställningen. Denna region följde vågfunktionen med betydande värden och kunde ändra storlek under simuleringen.En stråldelare konstruerades i form av en tunn potentialbarriär. Stråldelaren testades för att verifiera attde reflekterade och överförda vinklarna överensstämde med förväntningarna. En virtuell detektorkonstruerades och användes för kalibrering av stråldelaren för att bestämma vilken potential som resulterade i att vågpaketet delades in i två vågpaket med samma intensitet.En spegel med fast vinkel användes för att testa reflektionen av ett vågpaket för den reflekterade vinkeln och kom fram till att den överensstämde med förväntningarna för den. Att testa en tidsberoendespegel för olika frekvenser och amplituder utfördes med resultatet att den numeriska metoden kunde användas för att bestämma partiklarnas banor. Dessa resultat användes för att konstruera en större uppställning av ett experiment som var en liten delav Vaidmans föreslagna experiment. Dessa uppställningar gjordes i flera versioner. Alla uppställningar hade ett vågpaket som gick igenom en stråldelare och separerades i två vågpaket. Dessa två vågpaket reflekterades vid två speglar med olika frekvenser och interfererade sedan varandra antingen i en tom rymd eller vid en annan stråldelare. Resultatet av simuleringen av dessa inställningar var att partiklarnas banor kunde beräknas med ledande ekvation.

In this thesis, four different numerical techniques are implemented for the purpose of simulating resonant tunnelling diodes (RTDs). The chosen methods were: piecewise constant transfer matrix (TMM-C), piecewise linear transfer matrix (TMM-L), quantum transmitting boundary method (QTBM), and the Crank-Nicolson method (CN). The numerical methods converged compared with the known analytic simulation for plane waves tunnelling through a single barrier. To better represent the semiconductor-based RTDs, the effective mass approximation was adopted with accompanying modifications to the Hamiltonian operators to ensure the continuity of the wave function and its derivative. Using the Tsu-Esaki formula, the current density was calculated as a function of bias voltage for two different RTD devices. The numerically obtained current density was of the same order of magnitude as referenced experimental values but differed significantly enough to require better models if engineering applications are decided. The models in this thesis were able to display resonant tunnelling and a negative differential resistance (NDR), giving them plausible educational value.

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Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons are made between the coarse grid with refinement case and a fine grid without refinement case. Also, a non-negativity condition for two dimensions is derived to give a bound on the three major parameters: the diffusion coefficient and the spatial and time discretizations.

An unsteady, finite-rate, one-dimensional model has been developed for the analysis for gas-turbine combustors. The basis of the model is the one-dimensional, integral form of the conservation equations for multi-species, non-equilibrium, reacting mixtures. Special procedures were devised for the flow-division of the inlet flow into primary- and annular-flows, for both straight- and reverse-flow combustors. This allows the model to handle complete combustor configurations, which at present are beyond the reach of more sophisticated CFD tools. The model was validated with a steady-state analytical solution for a basic problem, and with steady-state results from a production code applied to a production combustor. Additional calculations show the ability of the code to predict blow-out due to rich and lean mixtures, and to predict the response of a combustor to perturbations in operating and boundary conditions. / Ph. D.

Many complexities arise when writing software for computational physics. The choice of underlying data structures, physics model representation, and numerical methods used for the solver all add to the overall complexity of a code and significantly affect the simulation speed and accuracy of the solution. This work has integrated multiple recently developed software tools into a unified framework called ELEMENTS. ELEMENTS contains tools to address the complexities of data representation and numerical methods implementation for computational physics applications. ELEMENTS consists of multiple software packages: Elements, MATAR, Swage, Geometry, and SLAM. MATAR is a performance portability and productivity implementation of data-oriented design that leverages KOKKOS for multi-architecture portability. MATAR's data-oriented design allows for highly efficient memory use through the use of contiguous memory allocation and access for optimal performance. The elements library contains the requisite mathematical functions for a wide range of numerical methods and high order field representation, including the Serendipity basis set that allows for a higher-order solution with fewer degrees of freedom than the more standard tensor product elements. Swage is a novel mesh class capable of representing all of the geometric entities required to implement low and high-order continuous and discontinuous Galerkin methods on unstructured hexahedral meshes as well as connectivity structures between the disparate index spaces. SLAM is a library for linear algebra solvers and tools for linking to external solver packages. Combining these tools allows for the research and development of novel methods for solving problems in computational physics. This work discusses the ELEMENTS package and reviews multiple numerical methods built using ELEMENTS.

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