PhD Defense at UZH Zurich. The talk summarize the main topics of the thesis: high order methods for ODEs, hyperbolic PDEs and model order reduction techniques.
Numerical simulations are extremely important to forecast physical events. In particular, this is true when experiments are too expensive or unfeasible. The field of numerical analysis studies how to obtain reliable simulations of physical phenomena. Physics provides the modeling equations, e. g. partial differential equations (PDEs), then numerical analysis creates numerical methods that approximate the solutions of such equations. In this manuscript, we focus on numerical methods for ordinary differential equations (ODEs) and hyperbolic PDEs.
ODEs can model many chemical and biological processes and the numerical methods to solve them are fundamental to solve also PDEs. Hyperbolic PDEs comprise many physical models, including fluid dynamics, transport equations, kinetic models and wave equations. The numerical methods for this kind of problems are vital for many engineering applications.
The schemes that we aim to obtain must verify many properties. They should converge to the analytical solution as the discretization scale decreases, they should be stable in order to produce spurious oscillations, they should guarantee a certain level of accuracy and they should be computable in reasonable times. Often, these last two factors are in contradiction as more accurate solutions require more computational time.
To tackle this problem we propose in this thesis some possible solutions. The first one is to speed up the convergence process by using high order accurate schemes. These schemes obtain much more accurate solutions with less refinements of the discretization scale with respect to low order accurate solutions. Hence, the computational costs needed to reach a certain error threshold is lower a priori. Another technique that we will use are implicit schemes. These schemes do not need to follow the restriction that explicit schemes have on the time discretization, allowing the use of less time steps. Finally, model order reduction techniques are tools that create a smaller discrete model, which represents, up to a certain error, an approximation of the solution manifold for parametric problems.
For high order accurate ODE solvers, we present in this work a class of arbitrarily high order schemes, called deferred correction (DeC) methods, which consist of an iterative procedure that, in a fixed number of loops, reaches an approximation of the required order. We study their A–stability for many possible orders of accuracy. In order to preserve positivity and conservation of physical quantities in production–destruction systems, we create a modified version of the DeC, which guarantees all these properties. This is possible thanks to the so–called Patankar trick, which makes the scheme linearly implicit. So far, the modified Patankar schemes were developed only up to third order of accuracy. The method we propose is arbitrarily high order accurate and unconditionally positivity preserving and conservative.
The rest of the thesis is focused on hyperbolic PDEs. We consider the residual distribution (RD) schemes as high order accurate spatial discretization technique in combination with the DeC for the time discretization. As a first step, we show a von Neumann stability analysis of the combination of these two methods, which suggests the optimal value of the stabilization parameters to maximize the time steps. This analysis uses Kreiss’ theorem as a tool to verify the stability of the family of matrices that evolve the Fourier coefficients of the solutions. The complications of this analysis are due to the different nature of different degrees of freedom inside the polynomial reconstruction.
Furthermore, we extend the RD DeC method to an implicit–explicit version for kinetic models. Kinetic models contain a source term that, in the asymptotic limit, becomes stiff. To deal with it, an implicit treatment of such a term is necessary. We propose an implicit—explicit RD DeC scheme that solves this type of models. Moreover, the proposed scheme is arbitrarily high order and asymptotic preserving, i. e., in the asymptotic regime the numerical solution converges to the analytical asymptotic limit. We prove these properties and we validate the theoretical results with numerical simulations.
Next, we study the model order reduction (MOR) algorithms for parametric hyperbolic problems. These techniques were originally developed for elliptic and parabolic problems and not all the algorithms can be extended to the hyperbolic framework. We propose an uncertainty quantification application of a MOR benchmark algorithm for hyperbolic problems. We show how the reduction can save computational time and we compute some statistical quantities, like mean and variance, of stochastic hyperbolic PDEs.
Finally, we extend this algorithm in order to gain more compression in the reduced model. Indeed, MOR algorithms are badly suited for advection dominated problems and most of the hyperbolic problems are of this kind. Even for the simplest wave transport problems, the classical MOR techniques fail to obtain a reasonable reduction, since they try to express the solution manifold as a linear combination of modes. What we propose in the last part of this thesis is to contextualize the PDEs into an arbitrary Lagrangian–Eulerian framework, which allows, through a transformation map, to align the advected features and to strongly compress the relevant information of the solution manifold. The transformation map must also be quickly computable in the reduced model and to do so, we use different regression techniques, such as polynomial regression and artificial neural networks, and we compare their performances.
All the algorithms and schemes are validated through adequate numerical simulations.