A higher‐order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions. Issue 2 (6th October 2022)

• Record Type: Journal Article
• Authors: Gupta, Aastha; Kaushik, Aditya; Sharma, Manju
• Is Part Of: Numerical methods for partial differential equations. Volume 39:Issue 2(2023)
• Page Start: 1220
• Page End: 1250
• Publication Date: 2022-10-06
• Subjects: boundary value -- grid equidistribution -- hybrid difference scheme -- initial‐boundary value problems -- mesh generation -- parameter uniform methods -- robust discretization -- singular perturbation; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22931
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin‐boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature‐based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first‐order in time and second‐order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.