Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation

We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy. Subsequently we extend these results to semi--lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings

Robust computational methods to simulate slow-fast dynamical systems governed by predator-prey models

Philosophiae Doctor - PhDNumerical approximations of multiscale problems of important applications in ecology are investigated. One of the class of models considered in this work are singularly perturbed (slow-fast) predator-prey systems which are characterized by the presence of a very small positive parameter representing the separation of time-scales between the fast and slow dynamics. Solution of such problems involve multiple scale phenomenon characterized by repeated switching of slow and fast motions, referred to as relaxationoscillations, which are typically challenging to approximate numerically. Granted with a priori knowledge, various time-stepping methods are developed within the framework of partitioning the full problem into fast and slow components, and then numerically treating each component differently according to their time-scales. Nonlinearities that arise as a result of the application of the implicit parts of such schemes are treated by using iterative algorithms, which are known for their superlinear convergence, such as the Jacobian-Free Newton-Krylov (JFNK) and the Anderson’s Acceleration (AA) fixed point methods

Discontinuous Galerkin Methods for Extended Hydrodynamics.

This dissertation presents a step towards high-order methods for continuum-transition flows. In order to achieve maximum accuracy and efficiency for numerical methods on a distorted mesh, it is desirable that both governing equations and corresponding numerical methods are in some sense compact. We argue our preference for a physical model described solely by first-order partial differential equations called hyperbolic-relaxation equations, and, among various numerical methods, for the discontinuous Galerkin method. Hyperbolic-relaxation equations can be generated as moments of the Boltzmann equation and can describe continuum-transition flows. Two challenging properties of hyperbolic-relaxation equations are the presence of a stiff source term, which drives the system towards equilibrium, and the accompanying change of eigenstructure. The first issue can be solved by an implicit treatment of the source term. To cope with the second difficulty, we develop a space-time discontinuous Galerkin method, based on Huynh’s “upwind moment scheme.” It is called the DG(1)–Hancock method. The DG(1)–Hancock method for one- and two-dimensional meshes is described, and Fourier analyses for both linear advection and linear hyperbolic-relaxation equations are conducted. The analyses show that the DG(1)–Hancock method is not only accurate but efficient in terms of turnaround time in comparison to other semiand fully discrete finite-volume and discontinuous Galerkin methods. Numerical tests confirm the analyses, and also show the properties are preserved for nonlinear equations; the efficiency is superior by an order of magnitude. Subsequently, discontinuous Galerkin and finite-volume spatial discretizations are applied to more practical equations, in particular, to the set of 10-moment equations, which are gas dynamics equations that include a full pressure/temperature tensor among the flow variables. Results for flow around a micro-airfoil are compared to experimental data and to solutions obtained with a Navier–Stokes code, and with particle-based methods. While numerical solutions in the continuum regime for both the 10-moment and Navier–Stokes equations are similar, clear differences are found in the continuum-transition regime, especially near the stagnation point, where the Navier–Stokes code, even when implemented with wall-slip, overestimates the density.Ph.D.Aerospace Engineering and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58411/4/ysuzuki_1.pd

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Exponential Integrator Methods for Nonlinear Fractional Reaction-diffusion Models

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this work, we propose an exponential integrator method for nonlinear fractional reaction-diffusion equations. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order FETD schemes. This numerical scheme combined with fractional centered differencing is used for simulating many important nonlinear fractional models in applications. We demonstrate the superiority of our method over competing second order FETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional centered differencing and matrix transfer technique in discretization of Riesz fractional derivatives. The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative. Furthermore, we present a preliminary result of the application of the M-L RDP approximation to develop a generalized exponetial integrator scheme for time-fractional nonlinear reaction-diffusion equation

High order semi-Lagrangian methods for BGK-type models in the kinetic theory of rarefied gases

In questa tesi vengono sviluppati e studiati schemi semilagrangiani innovativi di alto ordine per la soluzione numerica di equazioni cinetiche di tipo BGK, che descrivono il comportamento di un gas rarefatto. Si tratta di equazioni alle derivate parziali per l'evoluzione della funzione di distribuzione di un gas nello spazio delle fasi. I metodi di tipo semilagrangiano permettono di ricondurre il problema alla soluzione di equazioni differenziali ordinarie nella variabile temporale, in quanto l'evoluzione nello spazio avviene lungo le linee caratteristiche. Vengono proposti sostanziali miglioramenti dei metodi semi-Lagrangiani esistenti in letteratura per l'equazione BGK, insieme con alcune nuove applicazioni. Inizialmente si è cercato di ridurre il costo computazionale, dovuto principalmente alle tecniche di interpolazione. A questo scopo sono stati sviluppati metodi multi-step di tipo BDF (Backward Differentiation Formula) in alternativa a metodi di tipo Runge Kutta DIRK. Inoltre sono stati proposti e studiati particolari schemi semi-Lagrangiani che evitano completamente l'interpolazione spaziale. Nella tesi vengono sviluppati e studiati metodi numerici di ordine 1,2,3 per l'equazione BGK 1D in velocità. Mediante l'uso della riduzione di Chu, tali metodi sono stati estesi a domini fisici più realistici, 3D in velocità. Per i problemi ai limiti, sono state proposte tecniche originali per il trattamento di condizioni al contorno riflessive e diffusive. Infine, i metodi sono stati estesi a modelli cinetici di tipo BGK per miscele di gas inerti e reattive.This thesis focuses on high order shock capturing semi-Lagrangian methods for the numerical solutions of BGK-type equations. The methods are based on L-stable schemes for the solution of the BGK equations along the characteristics, and are asymptotic preserving, in the sense that they are able to capture the fluid dynamic limit. Two families of schemes are presented, which di ffer for the choice of the time integrator, Runge-Kutta or BDF, respectively. A further distinction concerns space discretization: some schemes are based on high order reconstruction, while others are constructed on the lattice in phase space, thus requiring no space interpolation. Numerical experiments show that schemes without interpolation can be cost-effective, especially for problems that do not require a fine mesh in velocity. In particular, BDF3 without interpolation appears to have the best performance in most tests. We investigated some applications and extensions to more physical interesting problems of the numerical schemes proposed for the one dimensional BGK equation. By means of the Chu reduction, we extended the schemes to more realistic physical domains, 3D in velocity. Moreover, we considered boundary-value problems and we studied the treatment of reflective and diffusive boundary conditions. Two numerical techniques to deal with diffusive boundary conditions are presented. The numerical results obtained are in good agreement with the ones present in literature. Finally, the methods have been extended to different BGK models for inert and reactive gas mixtures in three dimensional velocity space

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal