Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations

In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations, by means of tools from nonlinear control theory. Lyapunov-based and Small-Gain feedback stabilization methods are exploited and numerous illustrating applications are presented for systems with a globally asymptotically stable equilibrium point. The obtained results can be used for the control of the global discretization error as well.Comment: 33 pages, 9 figures. Submitted for possible publication to BIT Numerical Mathematic

Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems

In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge-Kutta (SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type. We present SRK methods composed of $L$ ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree $L$. Internal stability is maintained at large stage number through an ordering algorithm which limits internal amplification factors to $10 L^2$. Test results for mildly stiff nonlinear advection-diffusion-reaction problems with moderate ($\lesssim 1$) mesh P\'eclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.Comment: 20 pages, 7 figures, 3 table

Integrating-factor-based 2-additive Runge-Kutta methods for advection-reaction-diffusion equations

There are three distinct processes that are predominant in models of flowing media with interacting components: advection, reaction, and diffusion. Collectively, these processes are typically modelled with partial differential equations (PDEs) known as advection-reaction-diffusion (ARD) equations. To solve most PDEs in practice, approximation methods known as numerical methods are used. The method of lines is used to approximate PDEs with systems of ordinary differential equations (ODEs) by a process known as semi-discretization. ODEs are more readily analysed and benefit from well-developed numerical methods and software. Each term of an ODE that corresponds to one of the processes of an ARD equation benefits from particular mathematical properties in a numerical method. These properties are often mutually exclusive for many basic numerical methods. A limitation to the widespread use of more complex numerical methods is that the development of the appropriate software to provide comparisons to existing numerical methods is not straightforward. Scientific and numerical software is often inflexible, motivating the development of a class of software known as problem-solving environments (PSEs). Many existing PSEs such as Matlab have solvers for ODEs and PDEs but lack specific features, beyond a scripting language, to readily experiment with novel or existing solution methods. The PSE developed during the course of this thesis solves ODEs known as initial-value problems, where only the initial state is fully known. The PSE is used to assess the performance of new numerical methods for ODEs that integrate each term of a semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses object-oriented and software-engineering techniques to allow implementations of many existing and novel solution methods for ODEs with minimal effort spent on code modification and integration. The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK) method to solve the advection term of an ARD equation. A matrix exponential is used as the exponential function, but CFERK methods can use other numerical methods that model the flowing medium. The reaction term is solved separately using an explicit Runge-Kutta method because solving it along with the diffusion term can result in stepsize restrictions and hence inefficiency. The diffusion term is solved using a Runge-Kutta-Chebyshev method that takes advantage of the spatially symmetric nature of the diffusion process to avoid stepsize restrictions from a property known as stiffness. The resulting methods, known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy solutions in less computational time than competing methods for certain challenging semi-discretized ARD equations. This demonstrates the practical viability both of using CFERK methods for advection and a 3-splitting in general

Open issues in devising software for the numerical solution of implicit delay differential equations

AbstractWe consider initial value problems for systems of implicit delay differential equations of the formMy′(t)=f(t,y(t),y(α1(t,y(t))),…,y(αm(t,y(t)))),where M is a constant square matrix (with arbitrary rank) and αi(t,y(t))⩽t for all t and i.For a numerical treatment of this kind of problems, a software tool has been recently developed [6]; this code is called RADAR5 and is based on a suitable extension to delay equations of the 3-stage Radau IIA Runge–Kutta method.The aim of this work is that of illustrating some important topics which are being investigated in order to increase the efficiency of the code. They are mainly relevant to(i)the error control strategies in relation to derivative discontinuities arising in the solutions of delay equations;(ii)the integration of problems with unbounded delays (like the pantograph equation);(iii)the applications to problems with special structure (as those arising from spatial discretization of evolutions PDEs with delays).Several numerical examples will also be shown in order to illustrate some of the topics discussed in the paper

Stability of step size control based on a posteriori error estimates

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments

Forward, Tangent Linear, and Adjoint Runge Kutta Methods in KPP–2.2 for Efficient Chemical Kinetic Simulations

The Kinetic PreProcessor (KPP) is a widely used software environment which generates Fortran90, Fortran77, Matlab, or C code for the simulation of chemical kinetic systems. High computational efficiency is attained by exploiting the sparsity pattern of the Jacobian and Hessian. In this paper we report on the implementation of two new families of stiff numerical integrators in the new version 2.2 of KPP. One family is the fully implicit three-stage Runge Kutta methods, and the second family are singly diagonally-implicit Runge Kutta methods. For each family tangent linear models for direct decoupled sensitivity analysis, and adjoint models for adjoint sensitivity analysis of chemical kinetic systems are also implemented. To the best of our knowledge this work brings the first implementation of the direct decoupled sensitivity method and of the discrete adjoint sensitivity method with Runge Kutta methods. Numerical experiments with a chemical system used in atmospheric chemistry illustrate the power of the stiff Runge Kutta integrators and their tangent linear and discrete adjoint models. Through the integration with KPP–2.2. these numerical techniques become easily available to a wide community interested in the simulation of chemical kinetic systems

Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows

In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft