Numerical solution of index-2 differenial-algebraic equations

Práce se zabývá numerickým řešením algebraicko-diferenciálních rovnic. Tyto rovnice jsou nejprve popsány teoreticky a jsou ukázány jejich základní vlastnosti. Pozornost je věnována zejména indexu, jsou popsány nejpoužívanější indexy. Numerické řešení se zaměřuje na Hessenbergovy tvary algebraicko-diferenciálních rovnic indexu dva. Jsou zde odvozeny implicitní Runge-Kuttovy metody a metody zpětného derivování, které se používají pro řešení algebraicko-diferenciálních rovnic indexu 2.This bachelor´s thesis deals with numerical solution of differential-algebraic equations. At first these equations are described theoretically and their basic properties are presented. Main attention is paid to index and the most used indexes are described in details. Then the thesis concentrates on numerical solution of Hessenberg forms index-2 differential-algebraic equations. Implicit Runge-Kutta methods and backward differentiation formulas are derived. Those are used for solution of index-2 differential-algebraic equations.

Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method

This paper extends the algorithm of Benner, Heinkenschloss, Saak, and Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic Riccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142, doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.Comment: 21 pages, 2 figures, 4 table

High-order implicit time integration for unsteady incompressible flows

The spatial discretization of unsteady incompressible Navier–Stokes equations is stated as a system of differential algebraic equations, corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. Asymptotic stability of Runge–Kutta and Rosenbrock methods applied to the solution of the resulting index-2 differential algebraic equations system is analyzed. A critical comparison of Rosenbrock, semi-implicit, and fully implicit Runge–Kutta methods is performed in terms of order of convergence and stability. Numerical examples, considering a discontinuous Galerkin formulation with piecewise solenoidal approximation, demonstrate the applicability of the approaches and compare their performance with classical methods for incompressible flows

High-Order discontinuous Galerkin methods for incompressible flows

The spatial discretization of the unsteady incompressible Navier-Stokes equations is stated as a system of Differential Algebraic Equations (DAEs), corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. Runge-Kutta methods applied to the solution of the resulting index-2 DAE system are analyzed, allowing a critical comparison in terms of accuracy of semi-implicit and fully implicit Runge-Kutta methods. Numerical examples, considering a discontinuous Galerkin Interior Penalty Method with piecewise solenoidal approximations, demonstrate the applicability of the approach, and compare its performance with classical methods for incompressible flows.Peer ReviewedPostprint (published version

Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

[EN] In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and to use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.A.G.-Z. was supported by Secretaria Nacional de Ciencia y Tecnologia SENESCYT (Ecuador), through the scholarship "Becas de Fomento al Talento Humano", and Deutsche Forschungsgemeinschaft through Collaborative Research Centre Transregio. SFB TRR 154. P.F.-d.-C. was partially supported by grant no. RTI2018-102256-B-I00 (Spain). J.-C.C. acknowledges the support by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI), and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. V.M. was partially supported by Deutsche Forschungsgemeinschaft through the Excellence Cluster Math+ in Berlin, and Priority Program 1984 "Hybride und multimodale Energiesysteme: Systemtheoretische Methoden fur die Transformation und den Betrieb komplexer Netze".González-Zumba, A.; Fernández De Córdoba, P.; Cortés, J.; Mehrmann, V. (2020). Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems. Mathematics. 8(9):1-26. https://doi.org/10.3390/math8091393S12689Schein, O., & Denk, G. (1998). Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits. Journal of Computational and Applied Mathematics, 100(1), 77-92. doi:10.1016/s0377-0427(98)00138-1Winkler, R. (2004). Stochastic differential algebraic equations of index 1 and applications in circuit simulation. Journal of Computational and Applied Mathematics, 163(2), 435-463. doi:10.1016/j.cam.2003.12.017CONG, N. D., & THE, N. T. (2012). LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1. Stochastics and Dynamics, 12(04), 1250002. doi:10.1142/s0219493712500025Küpper, D., Kværnø, A., & Rößler, A. (2011). A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise. BIT Numerical Mathematics, 52(2), 437-455. doi:10.1007/s10543-011-0354-0Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 15(1), 9-20. doi:10.1007/bf02128236Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application. Meccanica, 15(1), 21-30. doi:10.1007/bf02128237Dieci, L., & Van Vleck, E. S. (2002). Lyapunov Spectral Intervals: Theory and Computation. SIAM Journal on Numerical Analysis, 40(2), 516-542. doi:10.1137/s0036142901392304Dieci, L., & Van Vleck, E. S. (2006). Lyapunov and Sacker–Sell Spectral Intervals. Journal of Dynamics and Differential Equations, 19(2), 265-293. doi:10.1007/s10884-006-9030-5Linh, V. H., & Mehrmann, V. (2009). Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations. Journal of Dynamics and Differential Equations, 21(1), 153-194. doi:10.1007/s10884-009-9128-7Linh, V. H., Mehrmann, V., & Van Vleck, E. S. (2010). QR methods and error analysis for computing Lyapunov and Sacker–Sell spectral intervals for linear differential-algebraic equations. Advances in Computational Mathematics, 35(2-4), 281-322. doi:10.1007/s10444-010-9156-1Dieci, L., Russell, R. D., & Van Vleck, E. S. (1997). On the Compuation of Lyapunov Exponents for Continuous Dynamical Systems. SIAM Journal on Numerical Analysis, 34(1), 402-423. doi:10.1137/s0036142993247311Talay, D. (1990). Second-order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochastics and Stochastic Reports, 29(1), 13-36. doi:10.1080/17442509008833606Dieci, L., Russell, R. D., & Van Vleck, E. S. (1994). Unitary Integrators and Applications to Continuous Orthonormalization Techniques. SIAM Journal on Numerical Analysis, 31(1), 261-281. doi:10.1137/0731014YU. RYAGIN, M., & RYASHKO, L. B. (2004). THE ANALYSIS OF THE STOCHASTICALLY FORCED PERIODIC ATTRACTORS FOR CHUA’S CIRCUIT. International Journal of Bifurcation and Chaos, 14(11), 3981-3987. doi:10.1142/s0218127404011600Definition and Classification of Power System Stability IEEE/CIGRE Joint Task Force on Stability Terms and Definitions. (2004). IEEE Transactions on Power Systems, 19(3), 1387-1401. doi:10.1109/tpwrs.2004.825981Verdejo, H., Vargas, L., & Kliemann, W. (2012). Stability of linear stochastic systems via Lyapunov exponents and applications to power systems. Applied Mathematics and Computation, 218(22), 11021-11032. doi:10.1016/j.amc.2012.04.063Verdejo, H., Escudero, W., Kliemann, W., Awerkin, A., Becker, C., & Vargas, L. (2016). Impact of wind power generation on a large scale power system using stochastic linear stability. Applied Mathematical Modelling, 40(17-18), 7977-7987. doi:10.1016/j.apm.2016.04.020Wadduwage, D. P., Wu, C. Q., & Annakkage, U. D. (2013). Power system transient stability analysis via the concept of Lyapunov Exponents. Electric Power Systems Research, 104, 183-192. doi:10.1016/j.epsr.2013.06.011Milano, F., & Zarate-Minano, R. (2013). A Systematic Method to Model Power Systems as Stochastic Differential Algebraic Equations. 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Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints

Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).In this thesis we study the initialization of multistep methods and parametrize some well-known classesof multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)multistep methods and parametric formulation of $\beta-$blocked multistep methods.Depending on the number of steps, a multistep method requires adequate number of initial values tostart the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.The proposed starters estimate the error by embedded methods.The second part concerns the variable step-size $\beta-$blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize $\beta-$blocked multistep methods forthe solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments. For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulationallows time adaptivity by construction

Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890