arXiv subjects

Gobinda Garai

Publications and source records attributed to Gobinda Garai.

7 recordsLinked to original sources

arXiv2026

Nonlinear Parareal-Incomplete OSWR Method for the Coupled Reaction-Diffusion System: Convergence Analysis and Computational Strategies

This paper presents a development of a nonlinear extension of the Parareal-Incomplete Optimized Schwarz Waveform Relaxation (OSWR) method, aimed at efficiently simulating coupled reaction-diffusion systems, with particular attention to those encountered in cardiac electrophysiology. The strategy leverages the synergy of spatial and temporal decomposition to tackle the computational challenges of large-scale, nonlinear simulations. The proposed approach is accompanied by a comprehensive convergence analysis. Extensive numerical experiments confirm the convergence of the nonlinear Parareal-Incomplete OSWR method. The results showcase its robust convergence behavior across various space-time subdomains, underscoring the method's reliability and effectiveness, especially in complex simulations involving multiple subdomains.

math.NA↗

arXiv2026

Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs

This paper introduces Exp-ParaDiag, a novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework. We develop and analyze Exp-ParaDiag based on first and second order accurate exponential integrators. We establish the convergence of the proposed methods both as preconditioned fixed-point iterations and as precon- ditioners within the GMRES framework. Furthermore, we extend the Exp-ParaDiag formulation to achieve sixth-order temporal accuracy using exponential integrators. The proposed approach is also generalized to nonlinear problems, for which convergence is rigorously demonstrated. A series of numerical experiments is presented to validate the theoretical results and to illustrate the robustness and efficiency of the developed methods.

math.NA↗

arXiv2023

Solution of the Optimal Control Problem for the Cahn-Hilliard Equation Using Finite Difference Approximation

This paper is concerned with the designing, analyzing and implementing linear and nonlinear discretization scheme for the distributed optimal control problem (OCP) with the Cahn-Hilliard (CH) equation as constrained. We propose three difference schemes to approximate and investigate the solution behaviour of the OCP for the CH equation. We present the convergence analysis of the proposed discretization. We verify our findings by presenting numerical experiments.

math.OC↗

arXiv2023

Diagonalization Based Parallel-in-Time Method for a Class of Fourth Order Time Dependent PDEs

In this paper, we design, analyze and implement efficient time parallel method for a class of fourth order time-dependent partial differential equations (PDEs), namely biharmonic heat equation, linearized Cahn-Hilliard (CH) equation and the nonlinear CH equation. We use diagonalization technique on all-at-once system to develop efficient iterative time parallel methods for investigating the solution behaviour of said equations. We present the convergence analysis of Parallel-in-Time (PinT) algorithms. We verify our findings by presenting numerical results.

math.NA↗

arXiv2023

Linear and Nonlinear Parareal Methods for the Cahn-Hilliard Equation

In this paper, we propose, analyze and implement efficient time parallel methods for the Cahn-Hilliard (CH) equation. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of the CH equation has. The CH equation generally needs to be simulated for a very long time to get the solution of phase coarsening stage. Therefore it is desirable to accelerate the computation using parallel method in time. We present linear and nonlinear Parareal methods for the CH equation depending on the choice of fine approximation. We illustrate our results by numerical experiments.

math.NA↗

arXiv2021

Convergence of substructuring Methods for the Cahn-Hilliard Equation

In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and since then it has appeared in many fields ranging from tumour growth simulation, image processing, thin liquid films, population dynamics etc. Being a non-linear equation, it is important to develop robust numerical techniques to solve the CH equation. Here we present the formulation of Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to CH equation and study their convergence behaviour. We consider the domain-decomposition based DN and NN methods in one and two space dimension for two subdomains and extend the study for multi-subdomain setting for NN method. We verify our findings with numerical results.

math.NA↗

arXiv2021

Convergence of the Neumann-Neumann Method for the Cahn-Hilliard Equation

In this paper, we analyze a substructuring type algorithm for the Cahn-Hilliard (CH) equation. Being a nonlinear equation, it is of great importance to develop efficient numerical schemes for investigating the solution behaviour of the CH equation. We present the formulation of Neumann-Neumann (NN) method applied to the CH equation and investigate the convergence behaviour of the same in one and two spatial dimension for two subdomains. We illustrate the theoretical results by providing numerical example.

math.NA↗