Convergence acceleration techniques for the iterative solution of system of equations arising in the discretisations of compressible flow problems governed by the steady state Euler or Navier-Stokes equations is considered. The system of PDE is discretised using a finite difference or finite volume method yielding a large sparse system of equations. A solution is computed by integrating the corresponding time dependent problem in time until steady state is reached. A convergence acceleration technique based on semicirculant approximations is applied. For scalar model problems, it is proved that the preconditioned coefficient matrix has a bounded spectrum well separated from the origin. A very simple time marching scheme such as the forward Euler method can be used, and the time step is not limited by a CFL-type criterion. Instead, the time step can asymptotically be chosen as a constant, independent of the number of grid points and the Reynolds number. Numerical experiments show that grid and parameter independent convergence is achieved also in more complicated problem settings. A comparison with a multigrid method shows that the semicirculant convergence acceleration technique is more efficient in terms of arithmetic complexity. Another convergence acceleration technique based on fundamental solutions is proposed. An algorithm based on Fourier technique is provided for the fast application. Scalar model problems are considered and a theory, where the preconditioner is represented as an integral operator is derived. Theory and numerical experiments show that for first order partial differential equations, grid independent convergence is achieved.

Because of the accuracy and ease of implementation, the Monte Carlo methodology is widely used in the analysis of nuclear systems. The estimated effective multiplication factor (keff) and flux distribution are statistical by their natures. In eigenvalue problems, however, neutron histories are not independent but are correlated through subsequent generations. Therefore, it is necessary to ensure that only the converged data are used for further analysis. Discarding a larger amount of initial histories would reduce the risk of contaminating the results by non-converged data, but increase the computational expense. This issue is amplified for large nuclear systems with slow convergence. One solution would be to use the convergence of keff or the flux distribution as the criterion for initiating accumulation of data. Although several approaches have been developed aimed at identifying convergence, these methods are not always reliable, especially for slow converging problems. This dissertation has attacked this difficulty by developing two independent but related methodologies. One aims to find a more reliable and robust way to assess convergence by statistically analyzing the local flux change. The other forms a basis to increase the convergence rate and thus reduce the computational expense. Eventually, these two topics will contribute to the ultimate goal of improving the reliability and efficiency of the Monte Carlo criticality calculations.

The one dimensional transient heat conduction equation was numerically modeled through matrix diagonalization and three time-discretization schemes. The discrete methods were first-order backward, second-order backward, and implicit finite difference schemes. All simulations used the central difference formula in the space dimension. Two relevant physical systems were considered: a uniformly conducting slab and a melting block of ice. The latter lead to a moving boundary system, or Stefan problem. The multiphysics of melting was numerically modeled through alternating updates of temperature and melt front profiles. Iterative simulations were run with regularly refined discretization meshes in both systems. In the case of the conducting slab, temperature at a fixed point in space and time was considered. For the Stefan problem, the melt front movement after a set time was the physical solution of interest. The accuracy of the convergent results was increased using Richardson acceleration and the Wynn's epsilon algorithm. Accuracy was improved for the moving boundary problem as well, but to a significantly lesser degree. The relative errors improved by five and two orders of magnitude for the conducting block and melting ice simulations, respectively. These relative errors were used to determine that matrix diagonalization is the most accurate numerical solution among the four considered. In both simulation convergence and acceleration potential, matrix diagonalization was superior to the implicit and explicit discretization solutions. However, matrix diagonalization required significantly more computational time. With the enhancement of convergence acceleration, the finite difference schemes obtained similar relative errors to the diagonalization model. This demonstrated the value of convergence acceleration in the classic dilemma for every programmer. There is always a balance struck between model sophistication, accuracy, and computational time. Convergence acceleration allows for a simpler numerical model to achieve comparable accuracy, and in less time than that required for sophisticated numerical models. The numerical models were also compared for stability through parameters that arose in each simulation. These parameters were the Courant-Friederichs-Lewy (CFL) condition and diagonalized eigenvalues. Though diagonalization was found to be the most accurate, it was determined that the backwards finite difference solutions are the easiest to evaluate for stability. In these solution methods, the CFL value allows the stability to be determined prior to running the simulation.

Computational fluid dynamics (CFD) is a significant tool routinely used indesign and optimization in aerospace industry. Often cases with unsteadyflows must be computed, and the long compute times of standard methods hasmotivated the present work on new implicit methods to replace the standardexplicit schemes. The implementation and numerical experiments were donewith the Swedish national flow solver Edge, developed by FOI,universities, and collaboration partners.The work is concentrated on a Lower-Upper Symmetric Gauss-Seidel (LU-SGS)type of time stepping. For the very anisotropic grids needed forReynolds-Averaged Navier-Stokes (RANS) computations of turbulent boundary layers,LU-SGS is combined with a line-implicit technique.  The inviscid flux Jacobians which contribute to the diagonalblocks of the system matrix are based on a flux splitting method with upwind type dissipation giving  control over diagonal dominance and artificial dissipation.The method is  controlled by several parameters, and comprehensivenumerical experiments were carried out to identify their influence andinteraction so that close to optimal values can be suggested. As an example,the optimal number of iterations carried out in a time-step increases with increased resolution of the computational grid.The numbering of the unknowns is important, and the numberings produced by mesh generators of Delaunay- and advancing front-type wereamong the best.The solver has been parallelized with the Message Passing Interface (MPI) for runs on multi-processor hardware,and its performance scales with the number of processors at least asefficiently as the explicit methods. The new method saves typicallybetween 50 and 80 percent of the runtime, depending on the case, andthe largest computations have reached 110M grid nodes. Theclassical multigrid acceleration for 3D RANS simulations was foundineffective in the cases tested in combination with the LU-SGS solverusing optimal parameters. Finally, preliminary time-accurate simulations for unsteady flows have shown promising results. / <p>QC 20141201</p>

This thesis introduces an alternative potentially low cost solution for hardware-in-the-loop (HIL) simulation based on the waveform relaxation (WR) method. The WR tech-nique is extended so that, without the need for a real-time simulator, the behaviour of an actual piece of physical hardware can nevertheless be tested as though it were connected to a large external electrical network. This is achieved by simulating the external network on an off-line electromagnetic transients (EMT) simulation program, and utilizing iterative exchange of waveforms between the simulation and the hardware by means of a spe-cialized Real-Time Player/Recorder (RTPR) interface device. The approach is referred to as waveform relaxation based hardware-in-the-loop (WR-HIL) simulation. To make the method possible, the thesis introduces several new innovations for stabi-lizing and accelerating the WR-HIL algorithm. It is shown that the classical WR shows poor or no convergence when at least one of the subsystems is an actual device. The noise and analog-digital converters’ quantization errors and other hardware disturbances can affect the waveforms and cause the WR to diverge. Therefore, the application of the WR method in performing HIL simulation is not straightforward and the classical WR need to be modified accordingly. Three convergence techniques are proposed to improve the WR-HIL simulation con-vergence. Each technique is evaluated by an experimental example. The stability of the WR-HIL simulation is studied and a stabilization technique is proposed to provide suffi-cient conditions for the simulation stability. The approach is also extended to include the optimization of the parameters of power system controllers located in geographically distant places. The WR-HIL simulation technique is presented with several examples. At the end of the thesis, suggestions for the future work are presented. / February 2016

Computational fluid dynamics (CFD) has become a significant tool routinely used in design and optimization in aerospace industry. Typical flows may be characterized by high-speed and compressible flow features and, in many cases, by massive flow separation and unsteadiness. Accurate and efficient numerical solution of time-dependent problems is hence required, and the efficiency of standard dual-time stepping methods used for unsteady flows in many CFD codes has been found inadequate for large-scale industrial problems. This has motivated the present work, in which major effort is made to replace the explicit relaxation methods with implicit time integration schemes. The CFD flow solver considered in this work is Edge, a node-based solver for unstructured grids based on a dual, edge-based formulation. Edge is the Swedish national CFD tool for computing compressible flow, used at the Swedish aircraft manufacturer SAAB, and developed at FOI, lately in collaboration with external national and international partners. The work is initially devoted to the implementation of an implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) type of relaxation in Edge with the purpose to speed up the convergence to steady state. The convergence of LU-SGS has been firstly accelerated by basing the implicit operator on a flux splitting method of matrix dissipation type. An increase of the diagonal dominance of the system matrix was the principal motivation. Then the code has been optimized by means of performance tools Intel Vtune and CrayPAT, improving the run time. It was found that the ordering of the unknowns significantly influences the convergence. Thus, different ordering techniques have been investigated. Finding the optimal ordering method is a very hard problem and the results obtained are mostly illustrative. Finally, to improve convergence speed on the stretched computational grids used for boundary layers LU-SGS has been combined with the line-implicit method. / QC 20120626