Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs

The chemical kinetics ODEs arising from operator-split reactive-flow simulations were solved on GPUs using explicit integration algorithms. Nonstiff chemical kinetics of a hydrogen oxidation mechanism (9 species and 38 irreversible reactions) were computed using the explicit fifth-order Runge-Kutta-Cash-Karp method, and the GPU-accelerated version performed faster than single- and six-core CPU versions by factors of 126 and 25, respectively, for 524,288 ODEs. Moderately stiff kinetics, represented with mechanisms for hydrogen/carbon-monoxide (13 species and 54 irreversible reactions) and methane (53 species and 634 irreversible reactions) oxidation, were computed using the stabilized explicit second-order Runge-Kutta-Chebyshev (RKC) algorithm. The GPU-based RKC implementation demonstrated an increase in performance of nearly 59 and 10 times, for problem sizes consisting of 262,144 ODEs and larger, than the single- and six-core CPU-based RKC algorithms using the hydrogen/carbon-monoxide mechanism. With the methane mechanism, RKC-GPU performed more than 65 and 11 times faster, for problem sizes consisting of 131,072 ODEs and larger, than the single- and six-core RKC-CPU versions, and up to 57 times faster than the six-core CPU-based implicit VODE algorithm on 65,536 ODEs. In the presence of more severe stiffness, such as ethylene oxidation (111 species and 1566 irreversible reactions), RKC-GPU performed more than 17 times faster than RKC-CPU on six cores for 32,768 ODEs and larger, and at best 4.5 times faster than VODE on six CPU cores for 65,536 ODEs. With a larger time step size, RKC-GPU performed at best 2.5 times slower than six-core VODE for 8192 ODEs and larger. Therefore, the need for developing new strategies for integrating stiff chemistry on GPUs was discussed.Comment: 27 pages, LaTeX; corrected typos in Appendix equations A.10 and A.1

Towards New High-Order Operator Splitting Time-Integration Methods

Operator splitting (OS) methods represent a powerful strategy to solve an extensive range of mathematical models in the form of differential equations. They have a long and celebrated history, having been successfully used for well over half a century to provide efficient numerical solutions to challenging problems. In fact, OS methods are often the only viable way to solve many problems in practice. The simplest, and perhaps, most well-known OS methods are Lie--Trotter--Godunov and the Strang--Marchuk methods. They compute a numerical solution that is first-, and second-order accurate in time, respectively. OS methods can be derived by imposing order conditions using the Campbell--Baker--Hausdorff formula. It follows that, by setting the appropriate order conditions, it is possible to derive OS methods of any desired order. An important observation regarding OS methods with order higher than two is that, according to the Sheng--Suzuki theorem, at least one of their defining coefficients must be negative. Therefore, the time integration with OS methods of order higher than two has not been considered suitable to solve deterministic parabolic problems, because the necessary backward time integration would cause instabilities. Throughout this thesis, we focus our attention on high-order (i.e., order higher than two) OS methods. We successfully assess the convergence properties of some higher-order OS methods on diffusion-reaction problems describing cardiac electrophysiology and on an advection-diffusion-reaction problem describing chemical combustion. Furthermore, we compare the efficiency performance of higher-order methods to second-order methods. For all the cases considered, we confirm an improved efficiency performance compared to methods of lower order. Next, we observe how, when using OS and Runge--Kutta type methods to advance the time integration, we can construct a unique extended Butcher tableau with a similar structure to the ones describing Generalized Additive Runge--Kutta (GARK) methods. We define a combination of methods to be OS-GARK methods. We apply linear stability analysis to OS-GARK methods; this allows us to conveniently analyze the stability properties of any combination of OS and Runge--Kutta methods. Doing so, we are able to perform an eigenvalue analysis to understand and improve numerically unstable solutions