Nonlinearity accompanied with stiffness in atmospheric boundary layer physical parameterizations is a well-known concern in numerical weather prediction (NWP) models. Nonlinear diffusion equations, furthermore, are a class of equations which are extensively applicable in different fields of science and engineering. Numerical stability and accuracy is a common concern in this class of equation. In the present research, a comprehensive effort has been made toward the temporal integration of such equations. The main goal is to find highly stable and accurate numerical methods which can be used specifically in atmospheric boundary layer simulations in weather and climate prediction models, and extensively in other models where nonlinear differential equations play an important role, such as magnetohydrodynamics and Navier-Stokes equations. A modified extended backward differentiation formula (ME BDF) scheme is adapted and proposed at the first stage of this research. Various aspects of this scheme, including stability properties, linear stability analysis, and numerical experiments, are studied with regard to applications for the time integration of commonly used nonlinear damping and diffusive systems in atmospheric boundary layer models. A new temporal filter which leads to significant improvement of numerical results is proposed. Nonlinear damping and diffusion in the turbulent mixing of the atmospheric boundary layer is dealt with in the next stage by using optimally stable singly-diagonally-implicit Runge-Kutta (SDIRK) methods, which have been proved to be effective and computationally efficient for the challenges mentioned in the literature. Numerical analyses are performed, and two schemes are modified to enhance their numerical features and stability. Three-stage third-order diagonally-implicit Runge-Kutta (DIRK) scheme is introduced by optimizing the error and linear stability analysis for the aforementioned nonlinear diffusive system. The new scheme is stable for a wide range of time steps and is able to resolve different diffusive systems with diagnostic turbulence closures, or prognostic ones with a diagnostic length scale, with enhanced accuracy and stability compared to current schemes. The procedure implemented in this study is quite general and can be used in other diffusive systems as well. As an extension of this study, high-order low-dissipation low-dispersion diagonally implicit Runge-Kutta schemes are analyzed and introduced, based on the optimization of amplification and phase errors for wave propagation, and various optimized schemes can be obtained. The new scheme shows no dissipation. It is illustrated mathematically and numerically that the new scheme preserves fourth-order accuracy. The numerical applications contain the wave equation with and without a stiff nonlinear source term. This shows that different optimized schemes can be investigated for the solution of systems where physical terms with different behaviours exist.

Finite volume methods employing second-order gradient reconstruction schemes are often utilized to computationally solve the governing equations of fluid mechanics and transport. These schemes, while not as dissipative as first-order schemes, frequently produce oscillatory solutions in regions of discontinuities and/or unsatisfactory levels of dissipation in smooth regions of the variable field. Limiters are often employed to reduce the inherent variable over- and under-shoot; however, they can significantly increase the numerical dissipation of a solution, eroding a scheme’s performance in smooth regions. A novel gradient reconstruction scheme, which shows significant improvement over traditional second-order schemes, is presented in this work. Two implementations of this Optimization-based Gradient REconstruction (OGRE) scheme are examined: minimizing an objective function based on the mismatch between local reconstructions at midpoints or selected quadrature points between cell stencil neighbors. Regardless of the implementation employed, the resulting gradient calculation is a compact, implicit method that can be used with unstructured meshes by employing an arbitrary computational stencil. An adjustable weighting parameter is included in the objective function that allows the scheme to be tuned towards either greater accuracy or greater stability. To address over- and undershoot of the variable field near discontinuities, non-local, non-monotonic (NLNM) and local, non-monotonic (LNM) limiters have also been developed, which operate by enforcing cell minima and maxima on dependent variable values projected to cell faces. The former determines minimum and maximum values for a cell through recursive reference to the minimum and maximum values of its upwind neighbors. The latter determines these bounding values through examination of the extrema of values of the dependent variable projected from the face-neighbor cell into the original cell. Steady state test cases on structured and unstructured grids are presented, exhibiting the low-dissipative nature of the scheme. Results are primarily compared to those produced by existing limited and unlimited second-order upwind (SOU) and first-order upwind (FOU). Solution accuracy, convergence rate and computational costs are examined.

This thesis compiles the publications I coauthored during my doctoral studies at University of Leipzig on the subject of optimizing thermodynamic systems, focusing on three optimization perspectives: maximum efficiency, maximum power, and maximum efficiency at given power. We considered two currently intensely studied models in finite-time thermodynamics, i.e., low-dissipation models and Brownian systems. The low-dissipation model is used to derive general bounds on the performance of real-world machines, while Brownian systems allow us to better understand the practical limits and features of small systems. First, we derived maximum efficiency at given power for various low-dissipation setups, with a particular focus on the behavior close to maximum power, which helps us to determine whether it is more beneficial to operate the system at maximum power, near maximum power or in a different regime. Then, we move to the design of maximum-efficiency and maximum-power protocols for Brownian systems under different boundary conditions. Particularly, when the constraints on control parameters are experimentally motivated, we presented a geometric method yielding maximum-efficiency and maximum-power protocols valid for systems with periodically scaled energy spectrum and otherwise arbitrary dynamics. Each chapter contains a short informal introduction to the matter as well as an outlook, pointing out the direction for our research in the future.

The goal of this thesis is to introduce the finite difference method (FDM) adjusted for usage in modeling of sound propagation, and other approaches that are used together with this method. These approaches include selective filtering and time integration using the Runge-Kutta method, which has low computer memory requirements. An important topic in modeling sound propagation are boundary conditions. The thesis examines and verifies several types of boundary conditions. Included in the thesis are solutions to example problems implemented in Matlab.