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Purpose: The purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time accurate, and useful for practical applications, have a significant limitation. They do not account for separable controllable numerical dissipation of the primary variables. Furthermore, they have little or no significant choices of altogether different algorithms that can be integrated in a single analysis to mitigate numerical oscillations that may occur. In lieu of such limitations, under the generalized single-step single-solve (GS4) umbrella, several of the deficiencies are circumvented. Design/methodology/approach: The GS4 framework encompasses a wide variety of LMS schemes that are all second-order time accurate and offers controllable numerical dissipation. Unlike existing state-of-art, the present framework permits implicit–implicit and implicit–explicit coupling of algorithms via differential algebraic equations (DAE). As further advancement, this study embeds proper orthogonal decomposition (POD) to further reduce model sizes. This study also uses an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to prescribed accuracy requirements. Simple linear/nonlinear transient numerical examples are presented to provide proof of concept. Findings: The present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration algorithms in altogether different subdomains in conjunction with the POD to advance and improve the computational efficiency. Originality/value: The novelty of this paper is the addition of reduced order modeling features, how it applies to the previous DAE-GS4 framework and the improvement of the computational efficiency. The proposed framework/tool kit provides all the needed flexibility, robustness and adaptability for engineering computations. [ABSTRACT FROM AUTHOR]

In this paper, an integrated study for measuring the solutions of the continuous and discrete Lyapunov equations is developed. A unified algebraic Lyapunov equation (UALE) will be given first. Then, by using a bilinear transformation, the mentioned equation can be transformed into the quasi-standard form of the discrete algebraic Lyapunov equation (DALE). Thus, those approaches appeared in the literature for solving the estimation problem of the solution of the Lyapunov equations may be used to solve the same problem for the UALE. By extending mainly the approach proposed in [20] for the continuous algebraic Lyapunov equation (CALE) associated with linear algebraic techniques, several upper and lower matrix bounds of the solution of the UALE are developed. Then, matrix bounds of the solutions of the CALE and DALE can be obtained directly by these obtained matrix bounds. Algorithms for getting better bounds of the solutions of the CALE and DALE are also given. By comparisons, the obtained solution bounds of the CALE and DALE are less restrictive or easier to be calculated than those parallel results appeared in the literature. Finally, numerical examples have given to demonstrate the merits of the proposed approach. [ABSTRACT FROM AUTHOR]