R. V. Zhalnin, M. S. Nefedov, S. K. Zinina

National Research Mordovia State University (Saransk, Russian Federation)

Abstract. This paper considers the Full Approximation Scheme (FAS) multigrid method for the Discontinuous Galerkin method with implicit time discretization. The objective of the research is to apply this method to efficient solution of problems governed by nonlinear partial differential equations. A computational algorithm has been developed that implements the Full Approximation Scheme multigrid method using Newton's method and an improved Newton-Krylov method to solve the arising nonlinear equations at each grid level of the multigrid method. This approach significantly improves the efficiency of the algorithm and reduces required computational resources. Numerical experiments were conducted applying both approaches for solving the Hopf equation. The influence of the regularization parameter and of the Courant number on the convergence rate of Newton's method outer iterations was investigated. It has been experimentally demonstrated that the use of the Newton-Krylov method significantly improves the overall performance of the computational process compared to the traditional Newton's method, although both approaches demonstrate a similar order of convergence, approaching second order when using quadratic basis functions.

Key Words: multigrid method, FAS, discontinuous Galerkin method, implicit time discretization, h-multigrid, p-multigrid, Newton-Krylov method

For citation: R. V. Zhalnin, M. S. Nefedov, S. K. Zinina. Application of the Full Approximation Scheme Multigrid Method to solving one-dimensional nonlinear partial differential equations by the Discontinuous Galerkin Method. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:4(2025), 435–450. DOI: https://doi.org/10.15507/2079-6900.27.202504.435-450

Information about the authors:

Ruslan V. Zhalnin, Ph.D. (Phys. and Math.), Dean of the Faculty of Mathematics and IT, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk, 430005, Russia), ORCID: https://orcid.org/0000-0002-1103-3321, zhrv@mrsu.ru

Mikhail S. Nefedov, Postgraduate Student, Department of Applied Mathematics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk, 430005, Russia), ORCID: https://orcid.org/0009-0002-7347-2191, snef7@yandex.ru

Svetlana K. Zinina, Ph.D. (Math.), Associate Professor, Department of Applied Mathematics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk, 430005, Russia), ORCID: https://orcid.org/0000-0003-3002-281X, zininaskh@math.mrsu.ru