ADAPTIVE GRID BASED FINITE DIFFERENCE METHODS FOR SOLUTION OF HYPERBOLIC PDES: APPLICATION TO COMPUTATIONAL MECHANICS AND UNCERTAINTY QUANTIFICATION

Novel finite-difference based numerical methods for solution of linear and nonlinear hyperbolic partial differential equations (PDEs) using adaptive grids are proposed in this dissertation. The overall goal of this research is to improve the accuracy and/or computational efficiency of numerical solutions via the use of adaptive grids and suitable modifications of a given low-order order finite-difference scheme. These methods can be grouped in two broad categories. The first category of adaptive FD methods proposed in the dissertation attempt to reduce the truncation error and/or enhance the accuracy of the underlying numerical schemes via grid distribution alone. Some approaches for grid distribution considered include those based on (i) a moving uniform mesh/domain, (ii) adaptive gradient based refinement (AGBR) and (iii) unit local Courant-Freidrich-Lewy (CFL) number. The improvement in the accuracy which is obtained using these adaptive methods is limited by the underlying scheme formal order of accuracy. In the second category, the CFL based approach proposed in the first category was extended further using defect correction in order to improve the formal order of accuracy and computational efficiency significantly (i.e. by at least one order or higher). The proposed methods in this category are constructed based upon the analysis of the leading order error terms in the modified differential equation associated with the underlying partial differential equation and finite difference scheme. The error terms corresponding to regular and irregular perturbations are identified and the leading order error terms associated with regular perturbations are eliminated using a non-iterative defect correction approach while the error terms associated with irregular perturbations are eliminated using grid adaptation. In the second category of methods involving defect correction (or reduction of leading order terms of truncation error), we explored two different approaches for selection of adaptive grids. These are based on (i) optimal grid dis- tribution and (ii) remapping with monotonicity preserving interpolation. While the first category of methods may be preferred in view of ease of implementation and lower computational complexity, the second category of methods may be preferred in view of greater accuracy and computational efficiency. The two broad categories of methods, which have been applied to problems involving both bounded and unbounded domains, were also extended to multidimensional cases using a dimensional splitting approaches. The performance of these methods was demonstrated using several example problems in computational uncertainty quantification (CUQ) and computational mechanics. The results of the application of the proposed approaches all indicate improvement in both the accuracy and computational efficiency (by about three orders of magnitude in some selected cases) of underlying schemes. In the context of CUQ, all three proposed adaptive finite different solvers are combined with the Gauss-quadrature sampling technique in excitation space to obtain statistical quantities of interest for dynamical systems with parametric uncertainties from the solution of Liouville equation, which is a linear hyperbolic PDE. The numerical results for four canonical UQ problems show both enhanced computational efficiency and improved accuracy of the proposed adaptive FD solution of the Liouville equation compared to its standard/fixed domain FD solutions. Moreover, the results for canonical test problems in computational mechanics indicate that the proposed approach for increasing the formal order of the underlying FD scheme can be easily implemented in multidimensional spaces and gives an oscillation-free numerical solution with a desired order of accuracy in a reasonable computational time. This approach is shown to provide a better computational time compared to both the underlying scheme (by about three orders of magnitude) and standard FD methods of the same order of accuracy

Adjoint-Based Error Estimation and Grid Adaptation for Functional Outputs from CFD Simulations

This study seeks to reduce the degree of uncertainty that often arises in computational fluid dynamics simulations about the computed accuracy of functional outputs. An error estimation methodology based on discrete adjoint sensitivity analysis is developed to provide a quantitative measure of the error in computed outputs. The developed procedure relates the local residual errors to the global error in output function via adjoint variables as weight functions. The three major steps in the error estimation methodology are: (1) development of adjoint sensitivity analysis capabilities; (2) development of an efficient error estimation procedure; (3) implementation of an output-based grid adaptive scheme. Each of these steps are investigated. For the first step, parallel discrete adjoint capabilities are developed for the variable Mach version of the U2NCLE flow solver. To compare and validate the implementation of adjoint solver, this study also develops direct sensitivity capabilities. A modification is proposed to the commonly used unstructured flux-limiters, specifically, those of Barth-Jespersen and Venkatakrishnan, to make them piecewise continuous and suitable for sensitivity analysis. A distributed-memory message-passing model is employed for the parallelization of sensitivity analysis solver and the consistency of linearization is demonstrated in sequential and parallel environments. In the second step, to compute the error estimates, the flow and adjoint solutions are prolongated from a coarse-mesh to a fine-mesh using the meshless Moving Least Squares (MLS) approximation. These error estimates are used as a correction to obtain highlyurate functional outputs and as adaptive indicators in an iterative grid adaptive scheme to enhance the accuracy of the chosen output to a prescribed tolerance. For the third step, an output-based adaptive strategy that takes into account the error in both the primal (flow) and dual (adjoint) solutions is implemented. A second adaptive strategy based on physics-based feature detection is implemented to compare and demonstrate the robustness and effectiveness of the output-based adaptive approach. As part of the study, a general-element unstructured mesh adaptor employing h-refinement is developed using Python and C++. Error estimation and grid adaptation results are presented for inviscid, laminar and turbulent flows

HIGH ACCURACY MULTISCALE MULTIGRID COMPUTATION FOR PARTIAL DIFFERENTIAL EQUATIONS

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

Refresher course in maths and a project on numerical modeling done in twos

These lecture notes accompany a refresher course in applied mathematics with a focus on numerical concepts (Part I), numerical linear algebra (Part II), numerical analysis, Fourier series and Fourier transforms (Part III), and differential equations (Part IV). Several numerical projects for group work are provided in Part V. In these projects, the tasks are threefold: mathematical modeling, algorithmic design, and implementation. Therein, it is important to draw interpretations of the obtained results and provide measures (Parts I-IV) how to build confidence into numerical findings such intuition, error analysis, convergence analysis, and comparison to manufactured solutions. Both authors have been jointly teaching over several years this class and bring in a unique mixture of their respective teaching and research fields

Krylov deferred correction methods for differential equations with algebraic constraints

In this dissertation, we introduce a new class of spectral time stepping methods for efficient and accurate solutions of ordinary differential equations (ODEs), differential algebraic equations (DAEs), and partial differential equations (PDEs). The methods are based on applying spectral deferred correction techniques as preconditioners to Picard integral collocation formulations, least squares based orthogonal polynomial approximations are computed using Gaussian type quadratures, and spectral integration is used instead of numerically unstable differentiation. For ODE problems, the resulting Krylov deferred correction (KDC) methods solve the preconditioned nonlinear system using Newton-Krylov schemes such as Newton-GMRES method. For PDE systems, method of lines transpose (MoLT ) couples the KDC techniques with fast elliptic equation solvers based on integral equation formulations and fast algorithms. Preliminary numerical results show that the new methods are of arbitrary order of accuracy, extremely stable, and very competitive with existing techniques, particularly when high precision is desired

Adaptive Numerical Methods for PDEs

This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods

Adaptive Finite Elements for Systems of PDEs: Software Concepts, Multi-level Techniques and Parallelization

In the recent past, the field of scientific computing has become of more and more importance for scientific as well as for industrial research, playing a comparable role as experiment and theory do. This success of computational methods in scientific and engineering research is next to the enormous improvement of computer hardware to a large extend due to contributions from applied mathematicians, who have developed algorithms which make real life applications feasible. Examples are adaptive methods, high order discretization, fast linear and non-linear solvers and multi-level methods. The application of these methods in a large class of problems demands for suitable and robust tools for a flexible and efficient implementation. In order to play a crucial role in scientific and engineering research, besides efficiency in the numerical solution, also efficiency in problem setup and interpretation of simulation results is of utmost importance. As modeling and computing comes closer together, efficient computational methods need to be applied to new sets of equations. The problems to be addressed by simulation methods become more and more complicated, ranging over different scales, interacting on different dimensions and combining different physics. Such problems need to be implemented in a short period of time, solved on complicated domains and visualized with respect to the demand of the user. %Only a modular abstract simulation environment will fulfill these requirements and allow to setup, solve and visualize real-world problems appropriately. In this work, the concepts and the design of the C++ finite element toolbox AMDiS (adaptive multidimensional simulations) are described. It is shown, how abstract data structures and modern software concepts can help to design user-friendly finite element software, which provides large flexibility in problem definition while on the other hand efficiently solves these problems. Also systems of coupled problems can be solved in an intuitive way. In order to demonstrate its possibilities, AMDiS has been applied to several non-standard problems. The most time-consuming part in most simulations is the solution of linear systems of equations. Multi-level methods use discretization hierarchies to solve these systems in a very efficient way. In AMDiS, such multi-level techniques are implemented in the context of adaptive finite elements. Several numerical results are given which compare this multigrid solver with classical iterative methods. Besides the development of more efficient algorithms also the growing hardware capabilities lead to an improvement of simulation possibilities. Modern computing clusters contain more and more processors and also personal computers today are often equipped with multi-core processors. In this work, a new parallelization approach has been developed which allows the parallelization of sequential code in a very easy way and reduces the communication overhead compared to classical parallelization concepts