Stability and Convergence Analysis of a Robust Numerical Scheme for the Variable Coefficient Multiterm Time‐Fractional Convection–Reaction–Diffusion Equation With Nonsmooth Solution

• Published in: Mathematical methods in the applied sciences Vol. 48; no. 11; pp. 11397 - 11418
• Main Authors: Roul, Pradip, Khandagale, Sameer
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.07.2025
• Subjects: Accuracy; compact difference method; Convection; Convergence; convergence analysis; Diffusion; graded mesh; Grid method; L2$$ {L}_2 $$‐norm error estimate; multiterm fractional convection–reaction–diffusion equation; Reaction-diffusion equations; Stability analysis
• ISSN: 0170-4214; 1099-1476; 1099-1476
• DOI: 10.1002/mma.10972

ABSTRACT In this paper, we propose a numerical scheme for approximating the solution of a variable coefficient multiterm Caputo time‐fractional convection–reaction–diffusion (MTCTFCRD) model, which generally exhibits a weak singularity at t=0$$ t=0 $$. The L1 technique is used to discretize the time‐fractional derivative (TFD) on a nonuniform mesh, while the space derivative is discretized on a uniform mesh using a fourth‐order compact finite difference (CFD) technique. The stability analysis of the method is introduced. The convergence analysis of the method is also rigorously investigated. Numerical examples are presented to corroborate in practice the accuracy and efficiency of the proposed technique and to validate the theoretical results as well. To justify the advantage of the suggested scheme, we compare the outcomes with those obtained by the uniform mesh technique. It is shown that the nonuniform mesh method attains an optimal‐order convergence in the temporal direction, whereas the uniform grid method fails to deliver an optimal order of convergence. The method is of fourth‐order accuracy in space. The computational times for the proposed numerical scheme have been provided.