scholarly journals A New Block Structural Index Reduction Approach for Large-Scale Differential Algebraic Equations

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2057
Author(s):  
Juan Tang ◽  
Yongsheng Rao
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Triangular Form ◽  
Large Scale Systems ◽  
Index Reduction ◽  
Dae Systems ◽  
New Generation ◽  
Block Structures

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications has emerged and became widely accepted; they generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAE systems with large dimensions, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on the Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameters has been presented. It leads to a block Pantelides’ algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples, and preliminary numerical experiments show that the time complexity of BPA can be reduced by at least O(ℓ) compared to the MPA, which is mainly consistent with the results of our analysis.

scholarly journals A New Block Structural Index Reduction Approach for Large-scale Differential Algebraic Equations

2020 ◽  
Author(s):  
Juan Tang ◽  
Yongsheng Rao
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Structural Index ◽  
Triangular Form ◽  
Large Scale Systems ◽  
Index Reduction ◽  
New Generation ◽  
Block Structures

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications emerged and became widely accepted, which generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAEs systems with large dimension, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on its Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameter has been presented.It leads to a block Pantelides’ algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples. And numerical experiments show that the time complexity of BPA can be reduced by at least O(ℓ) compared to the MPA, which is mainly consistent with the results of our analysis.

Structural Analysis Methods for Differential Algebraic Equations via Fixed-Point Iteration

2016 ◽  
Vol 13 (10) ◽  
pp. 7705-7711 ◽  
Author(s):  
Juan Tang ◽  
Wenyuan Wu ◽  
Xiaolin Qin ◽  
Yong Feng
Keyword(s):  
Fixed Point ◽  
Structural Analysis ◽  
Iteration Method ◽  
Large Scale ◽  
Complexity Analysis ◽  
Differential Algebraic Equations ◽  
Iteration Algorithm ◽  
Algebraic Equations ◽  
Fixed Point Iteration ◽  
Dae Systems

Motivated by Pryce’s structural analysis method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm (FPIA) and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method (BFPIM) which can be applied to immediately calculate the crucial canonical offsets for large-scale (coupled) DAE systems with block-triangular structure, and its complexity analysis is also given in this paper. Moreover, preliminary numerical experiments show that the time complexity of BFPIM can be reduced by at least O(l) compared to the FPIA.

scholarly journals Linear Differential-Algebraic Equations (Benchmark Proposal)

10.29007/4gj7 ◽  
2018 ◽  
Author(s):  
Patrick Musau ◽  
Diego Manzanas Lopez ◽  
Hoang-Dung Tran ◽  
Taylor T. Johnson
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Research Literature ◽  
Benchmark Problems ◽  
Cyber Physical System ◽  
Model Generation ◽  
Algebraic Equations ◽  
Dae Systems ◽  
Benchmark Suite ◽  
Linear Differential

This benchmark suite consists of eight small-to-large scale index-1 to index-3 linear differential algebraic systems (DAEs) derived from various application domains in engineering and science that exemplify the systemic prevalence of DAE systems in cyber-physical system applications. While in the last two decades numerous verification approaches and tools have been developed for systems described by ordinary differential equations, there is currently a lack of research methods for differential algebraic equations. Thus, the verification of DAE systems remains an open problem that has not been adequately addressed in the research literature. The following paper seeks to address this shortcoming by presenting a series of benchmark problems to stimulate the development of efficient and scalable tools for DAE verification and falsification. The benchmark models presented in this manuscript are available in the SpaceEx format using a tool named Daev for model generation.

Index reduction of differential algebraic equations by differential Dixon resultant

2018 ◽  
Vol 328 ◽  
pp. 189-202 ◽  
Author(s):  
Xiaolin Qin ◽  
Lu Yang ◽  
Yong Feng ◽  
Bernhard Bachmann ◽  
Peter Fritzson
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Index Reduction

scholarly journals Linear differential algebraic equations with constant coefficients

1970 ◽  
Vol 4 (2) ◽  
pp. 28-35 ◽  
Author(s):  
M Sahadet Hossain ◽  
M Mostafizur Rahman
Keyword(s):  
Existence And Uniqueness ◽  
Differential Algebraic Equations ◽  
Canonical Forms ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Constant Coefficients ◽  
Dae Systems ◽  
International University ◽  
Linear Differential ◽  
Modern Mathematics

Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are introduced. The canonical forms of DAEs are discussed widely to make them more efficient and easy for practical use. Also some numerical examples are discussed to clarify the existence and uniqueness of the system's solutions. Keywords: differential-algebraic equations, index concept, canonical forms. DOI: 10.3329/diujst.v4i2.4365 Daffodil International University Journal of Science and Technology Vol.4(2) 2009 pp.28-35

scholarly journals Approximation of potential-driven flow dynamics in large-scale self-similar tree networks

2011 ◽  
Vol 467 (2134) ◽  
pp. 2810-2824 ◽  
Author(s):  
Jason Mayes ◽  
Mihir Sen
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Large Scale ◽  
Differential Algebraic Equations ◽  
Flow Dynamics ◽  
Algebraic Equations ◽  
Tree Networks ◽  
Self Similar ◽  
Large Scale Flow ◽  
The Mathematical Model

Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced in complexity. For self-similar tree networks, this reduction can be made using the network’s structure in way that can allow simple, analytical solutions. For very large, but finite, networks, analytical solutions are more difficult to obtain. In the infinite limit, however, analysis is sometimes greatly simplified. It is shown that approximating large finite networks as infinite not only simplifies the analysis, but also provides an excellent approximate solution.

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