Second order algorithm with convergence analysis for viscoelastic flow model

• Published in: Journal of mathematics in industry Vol. 15; no. 1; pp. 11 - 20
• Main Authors: Nwaigwe, Chinedu, Micula, Sanda
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2025; Springer Nature B.V; SpringerOpen
• Subjects: Accuracy; Algorithms; Applications of Mathematics; Approximation; Convection; Convection-diffusion equation; Convergence; Convergence analysis; Diffusion coefficient; Dimensional analysis; Finite difference scheme; Fluid dynamics; Fluid flow; Fluids; Industrial applications; Investigations; Math. Appl. in Environmental Science; Mathematical and Computational Biology; Mathematical and Computational Engineering; Mathematical Methods in Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Method of manufactured solutions; Non-Newtonian fluids; Numerical analysis; Physics; Porous materials; Velocity; Viscoelastic fluids; Viscoelasticity; Viscosity; Finite difference scheme; 76A10; Method of manufactured solutions; 65M06; Viscoelastic fluids; Non-Newtonian fluids; 76M20; Convergence analysis; Convection-diffusion equation
• ISSN: 2190-5983; 2190-5983
• DOI: 10.1186/s13362-025-00176-x

Viscoelastic fluids have several applications in industry and technology, which has attracted a lot of investigation into their mathematical models. This paper presents the first-ever verifiable convergent second-order method, with complete theoretical analysis, for one-dimensional viscoelastic flow model. Existing studies have focused on first-order methods and without theoretical analysis. Here, the model is transformed into a form that involves the time derivative of spatial derivative. The convection and diffusion terms are discretized together by splitting the convection term into two parts and scaling with the diffusion coefficient, then combined with different terms of the diffusion approximation. The high-degree term is discretized with central difference spatial scheme and implicit-explicit time integrator, while the unsteady and external source terms are discretized implicitly. This yielded a positivity preserving numerical scheme which is second order accurate. We pose and prove several Lemmas and Theorems to establish consistency, boundedness and convergence of the method. Numerical experiments are then provided to verify the theoretical results, giving full confidence in the method. The results of this study provide a very useful tool for fluid dynamics simulations. The implication is that Researchers can now simulate 1D viscoelastic fluids, reliably, and without compromising some of the most important physics of the problem.