arXiv subjects

Deeksha Tomer

Publications and source records attributed to Deeksha Tomer.

5 recordsLinked to original sources

arXiv2026

Neumann-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay

Hyperbolic PDEs with time delay are essential for modeling a wide range of real-world applications. Since closed-form solutions are often not attainable, developing accurate and efficient numerical methods becomes crucial. In this work a novel substructuring waveform relaxation method namely Neumann-Neumann Waveform Relaxation (NNWR) has been implemented for numerically solving hyperbolic delay PDEs in the case of non-uniform subdomain partitioning. The convergence behavior is first analyzed using Fourier analysis to obtain an estimate, followed by a demonstration of finite-step convergence through Laplace transform techniques. Various test cases are presented as numerical illustrations.

math.NA↗

arXiv2026

Dirichlet-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay in Multiple Subdomains

Hyperbolic partial differential equations (PDEs) with time delay are essential mathematical tools used to model numerous physical systems where wave propagation or oscillations depend heavily on their historical states. As these applications scale in physical complexity, efficient parallel computing techniques become essential; however, developing highly scalable parallel solvers for delayed PDEs remains a significant computational challenge. To address this gap, this study extends the Dirichlet-Neumann Waveform Relaxation (DNWR) method to a multi-subdomain framework explicitly designed for solving time-delayed hyperbolic PDEs in parallel. We advance the underlying mathematical framework and provide a rigorous convergence analysis for both one-dimensional and two-dimensional spatial configurations. A central theoretical contribution of this work is the establishment of finite-step convergence for the multi-domain DNWR algorithm. These theoretical guarantees are firmly corroborated through comprehensive numerical experiments. Furthermore, a detailed comparative analysis against alternative domain decomposition techniques namely NNWR, OSWR, and Classical SWR, highlight the advantages of the proposed method. Finally, this approach provides a robust, highly parallelizable solver capable of efficiently handling complex hyperbolic PDEs with time delay.

math.NA↗

arXiv2026

Convergence of Substructuring Waveform Relaxation Algorithms for Hyperbolic PDEs with Time Delay

This article investigates the application and analysis of two substructuring waveform relaxation algorithms namely Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) for solving hyperbolic partial differential equations (PDEs) with time delay. These equations are relevant in numerous physical and engineering contexts, such as wave propagation, biological processes, and control systems, where the system's dynamics are influenced by past states. The study emphasizes the stability, convergence, and computational efficiency of these non-overlapping domain decomposition methods when applied to such problems. Specifically, the DNWR and NNWR algorithms are analyzed using both Fourier and Laplace transforms in asymmetric domain decomposition to assess their capability to manage delayed terms in hyperbolic systems. Using Fourier analysis, we establish linear convergence estimate for the numerical errors. Laplace transform analysis enables a more in-depth study for characterizing finite-step convergence. Additionally, we derive the optimal parameters required to achieve finite step convergence in presence of heterogeneous spatial domain. Theoretical findings are complemented by numerical experiments, showcasing the methods' effectiveness in maintaining accuracy while reducing computational complexity. Additionally, the study explores potential extensions to more complex problems and diverse applications.

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arXiv2026

Dirichlet-Neumann Waveform Relaxation Method with Multiple Subdomains for Reaction-Diffusion Equation with a Time Delay

In this study, we present the numerical investigation of the Dirichlet-Neumann Waveform Relaxation (DNWR) algorithm applied to multiple subdomains for the reaction-diffusion equation with time delay. Various arrangements of transmission conditions between subdomains are explored and a series of numerical experiments are conducted to evaluate and compare the efficiency and effectiveness of these configurations.

math.NA↗

arXiv2024

Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Methods for PDEs with Time Delay

We introduce and compare two domain decomposition based numerical methods, namely the Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation methods (DNWR and NNWR respectively), tailored for solving partial differential equations (PDEs) incorporating time delay. Time delay phenomena frequently arise in various real-world systems, making their accurate modeling and simulation crucial for understanding and prediction. We consider a series of model problems, ranging from Parabolic, Hyperbolic to Neutral PDEs with time delay and apply the iterative techniques DNWR and NNWR for solving in parallel. We present the theoretical foundations, numerical implementation, and comparative performance analysis of these two methods. Through numerical experiments and simulations, we explore their convergence properties, computational efficiency, and applicability to various types of PDEs with time delay.

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